ANALYTIC PROPERTIES OF MIRROR MAPS

C. KRATTENTHALER^{†} AND T. RIVOAL^{‡}

Dedicated to the memory of Alf van der Poorten

Abstract. We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [Math. Notes 71 (2002), 604–616]. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [Nucl. Phys. B359 (1991), 21–74] associated to the quintic hypersurface in ℙ^{4}(ℂ). In [Duke Math. J. 151 (2010), 175–218], we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in the Taylor expansions at 0 of these canonical coordinates are positive. Furthermore, we provide several results pertaining to the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address analytic continuation, points of singularity, and radius of convergence of these functions. We present several very precise conjectures on the radius of convergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

1. Introduction

In the focus of this article there is a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin [39] (to be defined below). This family contains many of the known one-variable mirror maps as special cases, including many of modular origin and the celebrated example of Candelas, de la Ossa, Green and Parkes [10] associated to the quintic hypersurface in ℙ^{4}(ℂ). For the geometric significance of these maps see [10, 26, 27, 36]. The number-theoretic properties of the coefficients in the Taylor expansions at 0 of these canonical coordinates and mirror maps have been investigated recently in [20] (cf. [13] for a far-reaching generalisation). Our aim here is to provide an as detailed as possible analysis of the analytic properties of canonical coordinates and mirror maps. Apart from the intrinsic interest in this kind of investigation, one motivation comes

Date: May 21, 2012. 2000 Mathematics Subject Classification. Primary 30B10; Secondary 11F11 14J32 30B40 33C20. Key words and phrases. mirror maps, canonical coordinates, analytic continuation, singular expansion,

generalised hypergeometric functions, modular forms.

†Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

‡Research partially supported by the project Q-DIFF, ANR-10-JCJC-0105, of the french “Agence Nationale de la Recherche”.

from the hope of finding more applications of the Diophantine method of uniformisation adélique simultanée of André [2], notably to non-modular situations.

Another connection between number theory and mirror maps can be found in the study of the arithmetic nature of values of the Riemann zeta function at integers. Apéry’s proof of the irrationality of ζ(3) (cf. [35]) was recast in terms of modular forms by Beukers [7]. The search for an extension of Beukers’ ideas to ζ(n), n ≥ 4, led Almkvist and Zudilin [1] to study systematically mirror maps associated to Fuchsian differential equations, not necessarily of hypergeometric type. In [21], the authors showed that, in fact, many of the examples of Beukers and of Almkvist and Zudilin can be obtained from suitable specialisations of hypergeometric multi-variable mirror maps. It would therefore be of great interest to extend the investigations undertaken in this paper addressing the analytic behaviour of the family of mirror maps to be introduced below to the family of multi-variable mirror maps in [21].

Let us now introduce this family of canonical coordinates and corresponding mirror maps. For a given integer N ≥ 1, let r_{1}, r_{2}, …, r_{d} denote the integers in {1, 2, …, N} which are coprime to N. It is well-known that d = φ(N), Euler’s totient function, which is given by φ(N) = N_{∏p|N}(1 − \frac{1}{p}). Set C_{N} := N^{φ(N)}_{∏p|N}p^{φ(N)/(p−1)}, which is an integer because p − 1 divides φ(N) for any prime p dividing N. Let us also define the Pochhammer symbol (α)_{m} for complex numbers α and non-negative integers m by (α)_{m} := α(α + 1) ⋯ (α + m − 1) if m ≥ 1, and (α)_{0} := 1. It can be proved (see [39, Lemma 1]) that, for any integer m ≥ 0,

B_{N}(m) := C_{N}^{m} ∏_{j=1}^{φ(N)} \frac{(r_{j}/N)_{m}}{m!}  (1.1)

is an integer.

Let us consider the hypergeometric differential operator L defined by

L := (z \frac{d}{dz} )^{φ(N_{1})+···+φ(N_{k})} − C_{N}z ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} (z \frac{d}{dz} + \frac{r_{i,j}}{N_{j}} ) .  (1.2)

Here, C_{N} = C_{N_{1}}C_{N_{2}} ⋯ C_{N_{k}} and the r_{i,j} ∈ {1, 2, …, N_{j}} form the residue classes modulo N_{j} which are coprime to N_{j}. Unless k = 1 and N = (2), the differential equation Ly = 0 is of order ≥ 2. We can construct two solutions as follows. Set H(x, m) := ∑_{n=0}^{m−1} \frac{1}{x+n} and

H_{N}(m) := ∑_{j=1}^{φ(N)} H(r_{j}/N, m) − φ(N)H(1, m).

Then, F_{N}(z) and G_{N}(z)+log(z)F_{N}(z) are two ℂ-linearly independent solutions to Ly = 0, where

F_{N}(z) := ∑_{m=0}^{∞} ( ∏_{j=1}^{k} B_{N_{j}}(m))z^{m}

and

G_{N}(z) := ∑_{m=1}^{∞} ( ∑_{j=1}^{k} H_{N_{j}}(m))( ∏_{j=1}^{k} B_{N_{j}}(m))z^{m},  (1.3)

and where log(z) denotes the principal branch of the logarithm. For simplicity, we write B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m) and H_{N}(m) := ∑_{j=1}^{k} H_{N_{j}}(m). Since B_{1}(m) = 1 and H_{1}(m) = 0 for all m ≥ 0, the series F_{N}(z) and G_{N}(z) do not change if one omits or adds components of 1 from/to N. We may therefore assume without loss of generality that N_{j} ≥ 2 for all j, which we shall do throughout the paper.

The power series F_{N}(z) and G_{N}(z) have radius of convergence 1/C_{N}. We prove in Section 3 that the functions F_{N}(z) and G_{N}(z)+log(z)F_{N}(z) can be analytically continued to ℂ \ [1/C_{N}, +∞) and ℂ \ ((−∞, 0] ∪ [1/C_{N}, +∞)), respectively.

Given the notation above, we define the canonical coordinate q_{N}(z) as the exponential of the quotient of the above two solutions, that is, by

q_{N}(z) := z exp(G_{N}(z)/F_{N}(z)).  (1.4)

Its compositional inverse, which we denote by z_{N}(q), is called (the corresponding) mirror map.

When k = 1 and N = (2), we have F_{(2)}(z) = (1 − 4z)^{−1/2}, which satisfies the differential equation (1 − 4z)y′ − 2y = 0. The function G_{(2)}(z) + log(z)F_{(2)}(z) defined formally by the above formula is not solution of that differential equation, but it turns out that all theorems stated below are still true in this case because

q_{(2)}(z) = (1 − \sqrt{1 − 4z})^{2}/(4z).  (1.5)

However, certain proofs do not work for this case, and we will say when.

The special case N = (5) has been of particular interest since it produces the earlier mentioned example of Candelas et al. [10].

In [20], we proved that, for any positive integers N_{1}, N_{2}, …, N_{k}, the canonical coordinate q_{N}(z) has integral Taylor coefficients. Our first result says that these coefficients are, in fact, positive. Its proof is given in Section 2. An essential ingredient there is a classical result of Kaluza [19] on the sign of coefficients in certain power series expansions (see Lemma 2.2).

Theorem 1.1. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2, all Taylor coefficients of q_{N}(z) at 0 are positive, except the constant coefficient, which vanishes.

A problem that suggests itself at this point is to find a combinatorial interpretation for the Taylor coefficients of q_{N}(z) or of z_{N}(q) (even if the latter may have negative coefficients, see Conjecture 1.8 below). Some progress in this direction can be found in [23].

The next theorem provides precise information on the radius of convergence and the asymptotic behaviour of the Taylor coefficients of the canonical coordinate q_{N}(z) as a power series in z. Here, and in the sequel, given N = (N_{1}, N_{2}, …, N_{k}), we employ the notation

Φ_{N} := ∑_{j=1}^{k} φ(N_{j}).  (1.6)

Theorem 1.2. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2, the following assertions hold:

(i) The radius of convergence of the Taylor series of q_{N}(z) is equal to 1/C_{N} and the Taylor series converges for any z such that |z| = 1/C_{N}.

(ii) The function q_{N}(z) has a singularity at z = 1/C_{N}.

(iii) For any z such that |z| ≤ 1/C_{N}, we have |q_{N}(z)| ≤ 1.

(iv) If Φ_{N} = 1, then the m-th Taylor coefficient of q_{(2)}(z) is equal to the m-th Catalan number \frac{1}{m+1} \binom{2m}{m}, m ≥ 1, and, hence, as m → ∞, it is equal to

\frac{4^{m}}{\sqrt{π}m^{3/2}} (1 + o(1)) .

(v) If Φ_{N} = 2, then, as m tends to ∞, the m-th Taylor coefficient of q_{N}(z) is equal to

const. × \frac{C_{N}^{m}}{m log^{2}(m)} (1 + o(1)) ,

where throughout the symbol “const.” stands for a non-zero constant. (vi) If Φ_{N} ≥ 3, then, as m tends to ∞, the m-th Taylor coefficient of q_{N}(z) is equal to

const. × \frac{C_{N}^{m}}{m^{Φ_{N}/2}} (1 + o(1)) .

In (iii), the inequality is always strict, except at z = 1/C_{N} when the series F_{N}(1/C_{N}) diverges to +∞, which happens in five cases: k = 1, N_{1} = 2, 3, 4 or 6, and k = 2, N_{1} = N_{2} = 2.

Theorem 1.2 is proved in Section 5. There, the analytic continuation of F_{N}(z) and G_{N}(z), which is discussed in Section 3, plays an important role, as well as the fine behaviour of these functions near the point 1/C_{N}, which is discussed in detail in Section 4.

By (1.4), the canonical coordinate q_{N}(z) is only defined for z in the disk of convergence of the series G_{N}(z) and F_{N}(z) involved in its definition. The knowledge of the analytic continuation of F_{N}(z) and G_{N}(z) from Section 3, combined with a theorem of Pólya [29] on zeroes of hypergeometric functions, allows us to show that q_{N}(z) can be continued to a function of the entire complex plane except for a branch cut. The corresponding proof is the subject of Section 6.

Theorem 1.3. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2, the following assertions hold: (i) The power series q_{N}(z) can be continued to an analytic function on ℂ\[1/C_{N}, +∞). (ii) The point 1/C_{N} is a branch point. (iii) We have

lim_{z→∞} q_{N}(z) = − exp ( − π cot(π/M_{N})),  (1.7)

where M_{N} = max(N_{1}, …, N_{k}), and where the limit has to be performed along a path that avoids the cut [1/C_{N}, +∞).

The monodromy of q_{N}(z) at z = 1/C_{N} follows from applying the well-known monodromy theory of solutions to (generalised) hypergeometric equations (cf. [8]) to the series F_{N}(z) and G_{N}(z) + log(z)F_{N}(z).

The function q_{N}(z), seen as a formal power series in z, is invertible at z = 0. Its formal inverse z_{N}(q) is a power series in q which converges in a neighbourhood of q = 0 (cf. [17, Theorems 2.4b, 2.4c]). The corresponding results for this compositional inverse z_{N}(q) that we are able to establish are less complete than Theorems 1.1–1.3 for q_{N}(z). First of all, we use Theorem 1.3 to get some information on the radius of convergence of z_{N}(q) as a power series in q.

Theorem 1.4. Let N_{1}, N_{2}, …, N_{k} be integers, all at least 2.

(i) The radius of convergence of the Taylor series of z_{N}(q) is at most exp ( − π cot(π/M_{N})).

(ii) If Φ_{N} ≥ 4, then the radius of convergence of the Taylor series of z_{N}(q) is at most q_{N}(1/C_{N}).

The proof of this theorem is found in Section 7.

Lemma 8.1 in Section 8 provides a comparison of the two values that feature in the above theorem in the case where Φ_{N} ≥ 4: namely, there it is shown that exp(−π cot(π/M_{N})) > q_{N}(1/C_{N}). As we shall see in Section 9, this inequality continues to hold for Φ_{N} = 3, but it is wrong for the remaining cases Φ_{N} = 1, 2.

In the cases of “small” Φ_{N}, we are able to provide precise information on the analytic continuation of z_{N}(q), see the theorem below. As we explain in Section 9, this information is in fact essentially available in the literature on the modular origin of the corresponding mirror maps. We point out that there exist several other hypergeometric functions (not covered by our series F_{N}(z)) which give rise to mirror maps of modular origin, see e.g. [34, 40]. It would be of interest to know whether there is a result analogous to the one below for these mirror maps.

Theorem 1.5. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2 with Φ_{N} ≤ 3, the radius of convergence of the Taylor series of z_{N}(q) is equal to exp ( − π cot(π/M_{N})). The function z_{N}(q) can be analytically continued to the unit disk with the exception of a set of poles or branch points which can be described precisely. In particular, − exp ( − π cot(π/M_{N})) is always a singularity of z_{N}(q). If N = (2), we have z_{N}(q) = q/(1 + q)^{2}. In all the eight other cases, the unit circle forms a natural boundary for z_{N}(q).

In Section 9, one also finds precise information on the poles, respectively branch points. In the case where Φ_{N} ≥ 4, we can offer only the partial result below concerning the

analytic nature of z_{N}(q), with the proof given in Section 10. There, and also later, we use the notion of a right slit neighbourhood of z_{0}, by which we mean a domain of the form (see Figure 1)

Δ(z_{0}; r, ε) := {z : |z − z_{0}| < r and | arg(z − z_{0})| > ε}  (1.8)

for some r, ε > 0, where it is understood that arg(z − z_{0}) is taken from (−π, π]. A left slit neighbourhood is defined in the obvious analogous way.

z_{0} r  ε

Figure 1. The right slit neighbourhood Δ(z_{0}; r, ε) of z_{0}

Proposition 1.6. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2 with Φ_{N} ≥ 4, the series z_{N}(q) can be analytically continued to a domain that contains the half-open segment [0, q_{N}(1/C_{N})). Moreover, this domain can be chosen so that it contains as well a right slit neighbourhood of q_{N}(1/C_{N}) (where the angle of the slit, controlled by the parameter ε in (1.8), can be chosen arbitrarily small).

Remark. The function q_{N}(z) can also be locally inverted at z = ∞. Consequently, the corresponding local inverse function z̃_{N}(q) is defined in a left slit neighbourhood of − exp(−π cot(π/M_{N})) (see the last paragraph in Section 10). The point − exp(−π cot(π/M_{N})) is always a singular point. Whether it is a pole, a branch point, or an essential singularity, this can be decided in each case by inspecting the expansions at ∞ for q_{N}(z) given in Lemma 4.1 (taking also into consideration the remark after the statement of the lemma), and by subsequently applying Lemma 4.6, with the accompanying remark in mind. Unfortunately, we do not know how to relate z̃_{N}(q) and z_{N}(q).

Certainly, the above result cannot even answer the innocent question of what the radius of convergence of z_{N}(q) as a power series in q is. However, we performed extensive calculations using the computer algebra system PARI/GP [28], in which we computed approximations to the radius of convergence using the quotient rule up to many digits. These computations provide abundant evidence for the exact value of the radius of convergence for the cases where Φ_{N} is “large.”

Conjecture 1.7. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2 with Φ_{N} ≥ 4, the radius of convergence of the Taylor series of z_{N}(q) is equal to q_{N}(1/C_{N}). Moreover, the point q_{N}(1/C_{N}) is the only singularity of z_{N}(q) on the boundary of its disk of convergence.

We discuss a possible line of argument to prove this conjecture in Section 11.

While, for Φ_{N} = 2, 3, Theorem 1.5 shows that the mirror map z_{N}(q) has a natural boundary, our computer experiments for Φ_{N} ≥ 4 did not allow us to develop an intuition whether or not there is a natural boundary for z_{N}(q) in these cases.

Concerning the Taylor coefficients of z_{N}(q) at q = 0, our numerical calculations suggest a very predictable behaviour.

Conjecture 1.8. Let N_{1}, N_{2}, …, N_{k} be positive integers, all at least 2.

(i) If Φ_{N} ≤ 3, the Taylor coefficients of z_{N}(q) have alternating signs.

(ii) If Φ_{N} = 4, the coefficients of q and of q^{3} in the Taylor series of z_{N}(q) are positive, while all other coefficients are negative, except the constant coefficient, which vanishes.

(iii) If Φ_{N} ≥ 5, the coefficient of q in the Taylor series of z_{N}(q) is positive, while all other coefficients are negative, except the constant coefficient, which vanishes.

We show in Section 12 that Conjecture 1.8 implies the first assertion in Conjecture 1.7. On the other hand, as we show in Section 13, Conjecture 1.7 implies a weak version of Conjecture 1.8.

There are three cases, where Conjecture 1.8.(i) is a theorem: in the case N = (2, 6), where z_{(2,6)}(q) = 1/j(τ) (with q = exp(2iπτ) and j(τ) the Dedekind–Klein j-invariant; see Section 9.3), this has been established by Asai, Kaneko and Ninomiya [4, Cor. 2, p. 94]. In the case N = (2, 2), alternance of coefficients follows immediately upon inspection of the explicit expression for z_{(2,2)}(q) given in (9.8). Similarly, in the case N = (2, 2, 2), alternance of coefficients follows immediately upon inspection of the explicit expression for z_{(2,2,2)}(q) given in (9.15). Moreover, Theorem 1.5, Lemma 4.2, Lemma 4.6, together with the standard theorems of singularity analysis (see [14, Ch. VI]) imply that Conjecture 1.8.(i) holds “asymptotically,” meaning that the coefficients of q^{m} of z_{N}(q) alternate in sign for all sufficiently large m.

The expression for q_{N}(z) in terms of F_{N}(z) and G_{N}(z) is not very convenient for a study of fine analytic properties of q_{N}(z) and of the corresponding mirror map z_{N}(q). The following conjecture, extending Theorem 1.1, is motivated by the search for an alternative expression, as we explain below.

Conjecture 1.9. For all integers N_{1}, N_{2}, …, N_{k} ≥ 2 and positive integers n, the coefficient of z^{m} in (C_{N}z − 1)^{n}q_{N}(z) is positive for all m ≥ n + 1.

Computer calculations indicate that, actually, the coefficients of z^{m} in (C_{N}z − 1)^{n}q_{N}(z) are alternating for m ≤ n+1 until the sign stabilises as described in the conjecture. That it must stabilise eventually follows from the singular expansion for q_{N}(z) given in Lemma 4.2 together with the standard theorems of singularity analysis (see [14, Ch. VI]). So the point here is that the sign stabilises already for m ≥ n + 1.

By a classical theorem of Hausdorff [16], Conjecture 1.9 implies that the sequence of coefficients of q_{N}(z) is a moment sequence for a finite measure dχ(t) on (0, C_{N}). As a consequence, q_{N}(z) could be written in the form

q_{N}(z) = z ∫_{0}^{C_{N}} \frac{dχ(t)}{1 − zt}.

By a theorem of Wirths [37], it would then follow that q_{N}(z) is univalent in the open half plane Re(z) < 1/C_{N}. In particular, this would lead to the following strengthening of Proposition 1.6: For all integers N_{1}, N_{2}, …, N_{k} ≥ 2 with Φ_{N} ≥ 4, the series z_{N}(q) can be analytically continued to a domain that contains the open segment (− exp(−π cot(π/M_{N})), q_{N}(1/C_{N})). Moreover, this domain can be chosen so that it contains as well a right slit neighbourhood of q_{N}(1/C_{N}) and a left slit neighbourhood of − exp(−π cot(π/M_{N})) (where the angle of the slit can be chosen arbitrarily small; cf. the last paragraph in Section 10).

From now on, in order to simplify notation, we let C := C_{N}, Φ := Φ_{N}, and M := M_{N}.

2. Proof of Theorem 1.1

A formal power series f(z) = ∑_{n=0}^{∞} f_{n}z^{n} ∈ ℂ[[z]] with f_{0} ≠ 0 is invertible in ℂ[[z]]. We can thus see f̂(z) = 1 − 1/f(z) as a formal power series. For the proof of Theorem 1.1, we need the following auxiliary result, whose proof can be found further down in this section.

Lemma 2.1. Let f(z) = ∑_{n=0}^{∞} f_{n}z^{n}, f_{0} = 1, be such that the Taylor coefficients of f̂(z) are non-negative. Let us consider g(z) = ∑_{n=0}^{∞} h_{n}f_{n}z^{n} where the sequence of real numbers (h_{n})_{n≥0} is non-decreasing and non-negative. Then the Taylor coefficients of g(z)/f(z) are non-negative.

If, in addition, all Taylor coefficients of f and f̂ are positive (except the constant coefficient of f̂) and the sequence (h_{n})_{n≥0} is strictly increasing, then the Taylor coefficients of g(z)/f(z) are positive, except the constant coefficient if h_{0} = 0.

To apply Lemma 2.1 to our situation, we need two further results. The first is a theorem due to Kaluza [19, Satz 3]. Strictly speaking, the assertion in Satz 3 in [19] deals only with the condition f_{n+1}f_{n−1} ≥ f_{n}^{2}; however, the strengthening given below is easily extracted from the proof in [19].

Lemma 2.2. Let f(z) = ∑_{n=0}^{∞} f_{n}z^{n}, f_{0} = 1, be such that f_{1} > 0 and f_{n+1}f_{n−1} ≥ f_{n}^{2} for all n ≥ 1. Then the Taylor coefficients of f̂(z) are non-negative.

If the stronger condition f_{n+1}f_{n−1} > f_{n}^{2} is satisfied for all n ≥ 1, then the Taylor coefficients of f̂(z) are positive (except the constant coefficient).

(It is easy to see that, if f̂(z) has non-negative Taylor coefficients, then f_{n} ≥ 0 for all n ≥ 0, but the converse is not true in general.)

Lemma 2.3. Let us fix the integers N_{1}, N_{2}, …, N_{k} ≥ 2. We have B_{N}(1) > 0 and

B_{N}(m + 1)B_{N}(m − 1) > B_{N}(m)^{2}

for all m ≥ 1. Furthermore, the sequence (H_{N}(m))_{m≥0} is positive and strictly increasing.

The proof of this lemma can be found at the end of this section. The combination of Lemmas 2.2 and 2.3 immediately implies the following corollary.

Corollary 2.4. Let N_{1}, N_{2}, …, N_{k} be positive integers, all at least 2. Then the Taylor coefficients of F̂_{N}(z) are positive (except the constant coefficient).

The above corollary shows that we can take f = F_{N} and g = G_{N} in Lemma 2.1. Hence, the power series G_{N}(z)/F_{N}(z) has positive Taylor coefficients (except the constant coefficient), a property which obviously remains true when we take the exponential. This proves Theorem 1.1.

Proof of Lemma 2.1. Let us write f̂(z) = ∑_{n=1}^{∞} f̂_{n}z^{n}. The relation f(z)(1 − f̂(z)) = 1 translates into

f_{n} − ∑_{k=1}^{n} f̂_{k}f_{n−k} = δ_{n,0}, for all n ≥ 0,  (2.1)

where δ_{n,0} is the Kronecker symbol. Furthermore, we have

\frac{g(z)}{f(z)} = g(z)(1 − f̂(z)) = ∑_{n=0}^{∞} z^{n}(h_{n}f_{n} − ∑_{k=1}^{n} f̂_{k}f_{n−k}h_{n−k}).

Since (h_{n})_{n≥0} is non-decreasing and non-negative, by (2.1) we have

h_{n}f_{n} − ∑_{k=1}^{n} f̂_{k}f_{n−k}h_{n−k} ≥ h_{n}f_{n} − ∑_{k=1}^{n} f̂_{k}f_{n−k}h_{n} = h_{n}(f_{n} − ∑_{k=1}^{n} f̂_{k}f_{n−k}) = h_{n}δ_{n,0} ≥ 0. (2.2)

The additional assertion in the lemma follows as well from the above arguments by observing that, because of the stronger assumptions, the first inequality in (2.2) is strict for n ≥ 1. □

Proof of Lemma 2.3. It is clear that B_{N}(m) > 0 for all m ≥ 0.

We want to prove that

\frac{B_{N}(m + 1)B_{N}(m − 1)}{B_{N}(m)^{2}} > 1  (2.3)

for all m ≥ 1. By definition, we have

\frac{B_{N}(m + 1)B_{N}(m − 1)}{B_{N}(m)^{2}} = ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{(r_{i,j}/N)_{m+1}(r_{i,j}/N)_{m−1}}{(r_{i,j}/N)_{m}^{2}} .

We observe that for every real number x > 0 and every integer m ≥ 1, we have

\frac{(x)_{m+1}(x)_{m−1}}{(x)_{m}^{2}} = \frac{x + m}{x + m − 1} > 1,

which immediately implies (2.3).

Concerning the second claim, we have

H_{N}(m) = ∑_{j=1}^{k} ( ∑_{i=1}^{φ(N_{j})} H(r_{i,j}/N, m) − φ(N_{j})H(1, m))

= ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (H(r_{i,j}/N_{j}, m) − H(1, m)),

from which we deduce that

H_{N}(m + 1) − H_{N}(m) = ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (\frac{1}{m + r_{i,j}/N_{j}} − \frac{1}{m + 1}) > 0,

because 0 < r_{i,j}/N_{j} < 1. Since H_{N}(0) = 0, we have proved that the sequence (H_{N}(m))_{m≥1} is positive and strictly increasing. □

3. Analytic continuation of F_{N}(z) and G_{N}(z)

For z complex with | arg(−z)| < π, let Log(z) denote the branch of the logarithm which assigns values with imaginary part between 0 and 2π. We shall continue to use log( .) for the principal branch of the logarithm. In order to facilitate the reading of the following paragraphs, as rule of thumb, below, whenever there appears z^{γ}, it has to be understood as z^{γ} = exp(γ Log(z)), whereas whenever there appears (−z)^{γ}, it has to be understood as (−z)^{γ} = exp(γ log(−z)).

For a real number h ≥ 0, set

F (h, z) := ∑_{n=0}^{∞} \frac{(α_{1} + h)_{n} ⋯ (α_{Φ} + h)_{n}}{(1 + h)_{n}^{Φ}} z^{n+h},  (3.1)

where Φ = Φ_{N} is given by (1.6) and the α_{j}’s run through the elements of the multiset (^{1})

{r_{i,j}/N_{j} : i = 1, …, φ(N_{j}), j = 1, …, k}.

We have

F (0, Cz) = F_{N}(z) and \frac{∂F}{∂h} (0, Cz) = Log(Cz)F_{N}(z) + G_{N}(z).  (3.2)

Generalised hypergeometric functions, such as the sum on the right-hand side of (3.1) have a Barnes-type integral representation, see [33, Sec. 4.6]. If we apply this to the right-hand side of (3.1) then, for any complex number z such that | arg(−z)| < π and any h ≥ 0, we obtain

∑_{n=0}^{∞} \frac{(α_{1} + h)_{n} ⋯ (α_{Φ} + h)_{n}}{(1 + h)_{n}^{Φ}} z^{n} =

− \frac{1}{2iπ} \frac{Γ(1 + h)^{Φ}}{Γ(α_{1} + h) ⋯ Γ(α_{Φ} + h)} ∫_{C} \frac{Γ(α_{1} + h + s) ⋯ Γ(α_{Φ} + h + s)}{Γ(1 + h + s)^{Φ}} \frac{π}{sin(πs)}(−z)^{s}ds, (3.3)

where C is a path from −i∞ to +i∞ such that 0, 1, 2, … lie on the right of C and the poles of the Γ(α_{ℓ} + h + s), ℓ = 1, …, Φ, lie to the left.

By multiplying both sides of (3.3) by z^{h}, and by using the relation Log(z) = log(−z)+iπ (recall the convention on the branches of the logarithm that we made in the first paragraph of this section), one obtains for | arg(−z)| < π and h ≥ 0 the equation

F (h, z) =

− \frac{e^{iπh}}{2iπ} \frac{Γ(1 + h)^{Φ}}{Γ(α_{1} + h) ⋯ Γ(α_{Φ} + h)} ∫_{C} \frac{Γ(α_{1} + h + s) ⋯ Γ(α_{Φ} + h + s)}{Γ(1 + h + s)^{Φ}} \frac{π}{sin(πs)}(−z)^{s+h}ds. (3.4)

1A multiset is a “set” where one allows repetitions of elements.

In particular, for h = 0 and z changed to Cz, this provides the analytic continuation of F_{N}(z) to the cut plane | arg(−z)| < π. Since F_{N}(z) is also analytic at any real point z ∈ [0, 1/C[, we get in this way the analytic continuation of F_{N}(z) to the cut plane | arg(1/C − z)| < π.

We now differentiate both sides of (3.4) with respect to h, and then set h = 0 and change z to Cz. After simplification, we get

G_{N}(z) = (Φψ(1) − ∑_{ℓ=1}^{Φ} ψ(α_{ℓ}))F_{N}(z)

− \frac{1}{2iπΓ(α_{1}) ⋯ Γ(α_{Φ})} ∫_{C} \frac{Γ(α_{1} + s) ⋯ Γ(α_{Φ} + s)}{Γ(1 + s)^{Φ}}( ∑_{ℓ=1}^{Φ} ψ(α_{ℓ} +s)−Φψ(1+s))\frac{π(−Cz)^{s}}{sin(πs)} ds, (3.5)

where ψ is the digamma function. A standard argument shows that the integral on the right-hand side of (3.5) is analytic in the cut plane | arg(−z)| < π, hence this is also the case for G_{N}(z). Then, from the series representation (1.3), we conclude that, in fact, G_{N}(z) is analytic in the cut plane | arg(1/C − z)| < π.

In [20, p. 216], we proved that

(Φψ(1) − ∑_{ℓ=1}^{Φ} ψ(α_{ℓ})) = ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (ψ(1) − ψ(r_{i,j}/N)) = log(C).  (3.6)

This quantity will reappear in the sequel.

We now prove the following result, which will be needed in the proof of Theorem 1.3.(iii) in Section 6.

Lemma 3.1. For integers N_{1}, …, N_{k} ≥ 2, let M = max(N_{1}, …, N_{k}), as before. Then we have

lim_{z→∞} (Log(z) + \frac{G_{N}(z)}{F_{N}(z)}) = −π cot(π/M) + iπ,

where the limit has to be performed along a path that avoids the cut [0, +∞). Here, F_{N}(z) and G_{N}(z) are given by their analytic continuations discussed just above, while Log(z) denotes the branch of the logarithm described at the beginning of this section.

Proof. By a well-known method, the integral (3.4) enables us to obtain an alternative expression for F (h, z) for |z| > 1/C in the cut plane | arg(1/C − z)| < π: we shift the contour C to the left, taking into account the various poles of the integrand coming from the product Γ(α_{1} + s) ⋯ Γ(α_{Φ} + s).

Let us start with the case k = 1, in which case α_{j} = r_{j}/N. Then, for |z| > 1/C and | arg(1/C − z)| < π, we have

F (h, z) = e^{iπh} ∑_{j=1}^{φ(N)} \frac{πΓ(1 + h)^{φ(N)}}{sin (π(\frac{r_{j}}{N} + h)) ∏_{ℓ=1}^{φ(N)} Γ(\frac{r_{ℓ}}{N} + h)} ∑_{ℓ=0}^{∞} \frac{∏_{ℓ=1,ℓ≠j}^{φ(N)} Γ(\frac{r_{ℓ}−r_{j}}{N} − ℓ)}{ℓ! Γ(1 − \frac{r_{j}}{N} − ℓ)^{φ(N)}}(−z)^{−ℓ−r_{j}/N},

(3.7)

and similarly

\frac{∂F}{∂h} (h, z)

= ∑_{j=1}^{φ(N)} \frac{∂}{∂h}(e^{iπh} \frac{πΓ(1 + h)^{φ(N)}}{sin (π(\frac{r_{j}}{N} + h)) ∏_{ℓ=1}^{φ(N)} Γ(\frac{r_{ℓ}}{N} + h)}) ∑_{ℓ=0}^{∞} \frac{∏_{ℓ=1,ℓ≠j}^{φ(N)} Γ(\frac{r_{ℓ}−r_{j}}{N} − ℓ)}{ℓ! Γ(1 − \frac{r_{j}}{N} − ℓ)^{φ(N)}}(−z)^{−ℓ−r_{j}/N}. (3.8)

In both cases, the leading term is the one corresponding to (−z)^{−1/N}, and thus

lim_{z→∞} \frac{\frac{∂F}{∂h} (0, z)}{F (0, z)} = log(C) − π cot(π/N) + iπ.

(Again, we use (3.6) to get the value log(C).) Using (3.2), the lemma follows in this case.

In the general case, it can be much more complicated to compute precisely the expansions because the poles might have multiplicity (i.e., some of the α_{j}’s might be equal or differ by an integer). The expected expansions are linear forms in the functions

(−z)^{−α_{ℓ}} log^{j}(−z)g_{ℓ,j}(1/z), j = 0, …, β_{ℓ} − 1,

with coefficients that depend on h. Here β_{ℓ} is the multiplicity of α_{ℓ}, and the g_{ℓ,j}(z)’s are holomorphic functions at z = 0. The main term in the expansions of F_{N}(z) and G_{N}(z) + Log(Cz)F_{N}(z) are those corresponding to (−z)^{−1/M} log^{β−1}(−z) (with the same maximal β in both cases), which can be computed without difficulty from (3.4). We get again

lim_{z→∞} \frac{\frac{∂F}{∂h} (0, z)}{F (0, z)} = log(C) − π cot(π/M) + iπ,

and the lemma follows. □

4. Singular expansions for q_{N}(z) and z_{N}(q)

The purpose of this section is to discuss the singular expansions of q_{N}(z) at z = ∞ (see Lemma 4.1) and z = 1/C_{N} (see Lemma 4.2), and of z_{N}(q) at the “corresponding” points q = − exp (−π cot(π/M_{N})) (recall (1.7)) and q = q_{N}(1/C_{N}). To obtain the latter, one has to combine Lemma 4.1 with Lemmas 4.3–4.5, respectively Lemma 4.2 with Lemma 4.6.

We start with the singular expansion of q_{N}(z) at z = ∞.

Lemma 4.1. Let N_{1}, N_{2}, …, N_{k} be positive integers, all of which at least 2, and let M = M_{N}, as before. Furthermore, let R be the least number different from 1/M in the multiset

ℜ = {r_{i,j}/N_{j} : i = 1, 2, …, φ(N_{j}), j = 1, 2, …, k}.

(i) If both 1/M and R appear exactly once in the multiset ℜ, then q_{N}(z) admits a singular expansion at ∞ of the form

q_{N}(z) = 𝔮_{0} + 𝔮_{1}(−z)^{−R+ \frac{1}{M}} + O((−z)^{−R+ \frac{1}{M} − \frac{1}{L}} log^{B_{3}−1}(−z)),  (4.1)

where 𝔮_{0} = − exp ( − π cot(π/M)), 𝔮_{1} is a non-zero constant, L = lcm(N_{1}, N_{2}, …, N_{k}), and B_{3} is the multiplicity of the third-smallest element in ℜ.

(ii) If 1/M appears exactly once in the multiset ℜ, while R appears with multiplicity B_{2}, then q_{N}(z) admits a singular expansion at ∞ of the form

q_{N}(z) = 𝔮_{0} + 𝔮_{1}(−z)^{−R+ \frac{1}{M}} log^{B_{2}−1}(−z) + O((−z)^{−R+ \frac{1}{M}} log^{B_{2}−2}(−z)),  (4.2)

where 𝔮_{0} and 𝔮_{1} are the same constants as in (i). (iii) If 1/M appears with multiplicity at least 2 in ℜ, then q_{N}(z) admits a singular

expansion at ∞ of the form

q_{N}(z) = 𝔮_{0} + 𝔮̃_{1} log^{−1}(−z) + O( log^{−2}(−z)),  (4.3)

where 𝔮_{0} has the same meaning as in (i), and 𝔮̃_{1} is a non-zero constant.

Remark. If the multiset ℜ is in fact a set, i.e., if all elements of ℜ appear with multiplicity 1, then q_{N}(z) admits a Puiseux expansion in (−z)^{−1/L}, of which (4.1) shows the first terms. In all other cases, the singular expansion at z = ∞ has terms containing log(−z).

Proof of Lemma 4.1. (i) By (3.2) and (in case B_{3} ≥ 2: the appropriately generalised) expansions (3.7) and (3.8), we know that

q_{N}(z) = \frac{1}{C} exp ( \frac{\frac{∂F}{∂h} (0, Cz)}{F (0, Cz)}),  (4.4)

where

F (0, Cz) = F_{1}(−Cz)^{− \frac{1}{M}} + F_{2}(−Cz)^{−R} + O((−z)^{−R− \frac{1}{L}} log^{B_{3}−1}(−z))  (4.5)

and

\frac{∂F}{∂h} (0, Cz) = G_{1}(−Cz)^{− \frac{1}{M}} + G_{2}(−Cz)^{−R} + O((−z)^{−R− \frac{1}{L}} log^{B_{3}−1}(−z)).  (4.6)

Here, F_{1}, F_{2}, G_{1}, G_{2} are explicit non-zero constants. If we use this in (4.4), then we obtain

q_{N}(z) = \frac{1}{C} exp (\frac{G_{1}}{F_{1}} (1 + (\frac{G_{2}}{G_{1}} − \frac{F_{2}}{F_{1}} ) (−Cz)^{−R+ \frac{1}{M}} + O((−z)^{−R+ \frac{1}{M} − \frac{1}{L}} log^{B_{3}−1}(−z)))) .

From the explicit expressions and (3.6), it is not difficult to see that G_{1}/F_{1} = log(C) − π cot(π/M) + iπ and that, furthermore, \frac{G_{2}}{G_{1}} ≠ \frac{F_{2}}{F_{1}} . The assertions in (i) now follow easily upon expansion of the exponential.

(ii) We proceed in the same way as in (i). Here, the expansions (4.5) and (4.6) must be replaced by

F (0, Cz) = F_{1}(−Cz)^{− \frac{1}{M}} + F_{2}(−Cz)^{−R} log^{B_{2}−1}(−z) + O((−z)^{−R} log^{B_{2}−2}(−z))

and

\frac{∂F}{∂h} (0, Cz) = G_{1}(−Cz)^{− \frac{1}{M}} + G_{2}(−Cz)^{−R} log^{B_{2}−1}(−z) + O((−z)^{−R} log^{B_{2}−2}(−z)).

The constants F_{1}, F_{2}, G_{1}, G_{2} are the same as in (i). The remaining steps are completely analogous to those in (i) and are therefore omitted.

(iii) Again, we proceed in the same way as in (i). Here, the expansions (4.5) and (4.6) must be replaced by

F (0, Cz) = F_{1}(−Cz)^{− \frac{1}{M}} log^{B_{1}−1}(−z) + F̃_{2}(−Cz)^{− \frac{1}{M}} log^{B_{1}−2}(−z)

+ O((−z)^{− \frac{1}{M}} log^{B_{2}−3}(−z))

and

\frac{∂F}{∂h} (0, Cz) = G_{1}(−Cz)^{− \frac{1}{M}} log^{B_{1}−1}(−z) + G̃_{2}(−Cz)^{− \frac{1}{M}} log^{B_{1}−2}(−z)

+ O((−z)^{− \frac{1}{M}} log^{B_{1}−3}(−z)).

The constants F_{1}, F_{2} are the same as in (i). The remaining steps are completely analogous to those in (i) and are therefore omitted. □

The next lemma addresses the singular expansion of q_{N}(z) at z = 1/C_{N}.

Lemma 4.2. Let N_{1}, N_{2}, …, N_{k} be positive integers, all of which at least 2, and let C = C_{N}, Φ = Φ_{N}, as before.

(i) If Φ = 2, then q_{N}(z) admits a singular expansion at 1/C of the form

q_{N}(z) = 1 + q_{1} log^{−1}(1 − Cz) + O( log^{−2}(1 − Cz)),  (4.7)

where q_{1} > 0. (ii) If Φ ≥ 3 is odd, then q_{N}(z) admits a singular expansion at 1/C of the form

q_{N}(z) = q_{N}(1/C) + q_{1}(1 − Cz) + q_{2}(1 − Cz)^{2} + ⋯

+ q_{d}(1 − Cz)^{d} + q_{d+ \frac{1}{2}} (1 − Cz)^{d+ \frac{1}{2}} + O((1 − Cz)^{d+1}), (4.8)

where d = \frac{Φ−3}{2}, q_{1} < 0, and (−1)^{d+1}q_{d+ \frac{1}{2}} > 0. (iii) If Φ ≥ 4 is even, then q_{N}(z) admits a singular expansion at 1/C of the form

q_{N}(z) = q_{N}(1/C) + q_{1}(1 − Cz) + q_{2}(1 − Cz)^{2} + ⋯

+ q_{d−1}(1 − Cz)^{d−1} + q_{d+}(1 − Cz)^{d} log(1 − Cz) + O((1 − Cz)^{d}), (4.9)

where d = \frac{Φ−2}{2}, q_{1} < 0, and (−1)^{d+1}q_{d+} > 0.

Proof. We proceed by using the theory of hypergeometric differential equations to determine the form of the singular expansion of the quotient G_{N}(z)/F_{N}(z), see (4.19). This is then translated in the final step into the claimed singular expansions for q_{N}(z).

First of all, from Section 3 we know that F_{N}(z) and G_{N}(z) + log(az)F_{N}(z) (for any a ≠ 0) can be analytically continued to ℂ \ [1/C, +∞) and ℂ \ ((−∞, 0] ∪ [1/C, +∞)), respectively. We want to determine their behaviour around the point z = 1/C. The exponents at the regular singular point z = 1/C of the differential equation Ly = 0 (with L being defined in (1.2)) are 0, 1, …, Φ − 2, and

(Φ − 1) − ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} \frac{r_{i,j}}{N_{j}} .  (4.10)

By the elementary identity

∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} \frac{r_{i,j}}{N_{j}} = \frac{Φ}{2},  (4.11)

the value (4.10) simplifies to \frac{Φ}{2} − 1. By the theory of hypergeometric differential equations (cf. [11, Ch. 4, Sec. 8]), a basis over ℂ of solutions of L consists of Φ − 1 functions f_{1}(z), …, f_{Φ−1}(z) holomorphic at z = 1/C, together with another solution f_{Φ}(z) which can be described as follows:

a) if Φ is odd, then f_{Φ}(z) = (1 − Cz)^{Φ/2−1}u(z), where u(z) is holomorphic at z = 1/C;

b) if Φ is even, then f_{Φ}(z) = v(z) + (1 − Cz)^{Φ/2−1} log(1 − Cz)u(z), where both u(z)

and v(z) are holomorphic at z = 1/C. It follows that, in a neighbourhood of 1/C avoiding the cut [1/C, +∞), we have (^{2})

F_{N}(z) = f(z) + (1 − Cz)^{Φ/2−1}L(z)g(z)  (4.12)
G_{N}(z) = f̆(z) + (1 − Cz)^{Φ/2−1}L(z)ğ(z),  (4.13)

where f, f̆, g and ğ are holomorphic around z = 1/C and L(z) = 1 if Φ is odd, respectively L(z) = − log(1 − Cz) if Φ is even.

Concerning the coefficients of F_{N}(z), by Stirling’s formula, we have

B_{N}(m) = ( ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{1}{Γ(r_{i,j}/N_{j})}) · \frac{C^{m}}{m^{Φ/2}} (1 + o(1)) ,  m → ∞,  (4.14)

which implies that g(1/C) > 0 by the classical link between singularities of an analytic function h and the asymptotic behaviour of the Taylor coefficients of h when these are positive (see [14, Ch. VI]).

2Since the function G_{N}(z) + log(az)F_{N}(z) (with a ≠ 0) is a solution of Ly = 0, it can be written in a form similar to the right-hand side of (4.12). Application of (4.12) to log(az)F_{N}(z) then gives (4.13)

because log(az) is holomorphic at z = 1/C.

Concerning the coefficients of G_{N}(z), we have

B_{N}(m)H_{N}(m) = B_{N}(m)( log(C) − \frac{Φ}{2m} + O(\frac{1}{m^{2}} )).  (4.15)

(In fact, log(C) appears under the form ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (ψ(1)−ψ(r_{i,j}/N)), see (3.6).) Hence, using (4.15), we can make (4.13) more precise (^{3}):

G_{N}(z) = log(C)F_{N}(z) + f̃(z) + (1 − Cz)^{Φ/2}L(z)g̃(z),  (4.16)

where f̃ and g̃ are holomorphic at z = 1/C, and g̃(1/C) ≠ 0.

In order to proceed, we need the following auxiliary result.

For any vector N of positive integers, the limit

S := lim_{z→1/C} (G_{N}(z) − log(C)F_{N}(z))  (4.17)

exists, is finite and is < 0. Furthermore, it is equal to f̃(1/C). Above and in the sequel, the limit z → 1/C is understood along real numbers z < 1/C. In order to see (4.17), we observe that, for |z| < 1/C, we have

G_{N}(z) − log(C)F_{N}(z) = ∑_{m=0}^{∞} B_{N}(m)(H_{N}(m) − log(C))z^{m}.

The series on the right-hand side converges for z = 1/C because

|B_{N}(m)(H_{N}(m) − log(C))| = O (\frac{C^{m}}{m^{Φ/2+1}} )

and Φ/2 + 1 > 1. By Abel’s theorem, the limit S in (4.17) exists and

S = ∑_{m=0}^{∞} B_{N}(m)(H_{N}(m) − log(C))C^{−m},

the right-hand side being finite.

Secondly, since H(x, n) = ψ(n + x) − ψ(x), it is easy to see that

H_{N}(m) − log(C) = ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (ψ(m + \frac{r_{i,j}}{N_{j}} ) − ψ(m + 1)),

where we used (3.6) again. Since the function ψ is strictly increasing on (0, +∞), and since 0 < \frac{r_{i,j}}{N} < 1, we deduce that

H_{N}(m) − log(C) < 0 for all m ≥ 0.  (4.18)

Hence S < 0.

Finally, since Φ/2 > 0, Eq. (4.16) implies that

lim_{z→1/C} (G_{N}(z) − log(C)F_{N}(z)) = lim_{z→1/C} (f̃(z) + (1 − Cz)^{Φ/2}L(z)g̃(z)) = f̃(1/C), thus completing the proof of (4.17).

3This is more precise when one transforms log(C)F_{N}(z) using (4.12).

We may now continue with the proof of the lemma. By the remarks preceding (4.17), we have

\frac{G_{N}(z)}{F_{N}(z)} = log(C) + \frac{f̃(z) + (1 − Cz)^{Φ/2}L(z)g̃(z)}{f(z) + (1 − Cz)^{Φ/2−1}L(z)g(z)}.  (4.19)

This is now translated to q_{N}(z) = z exp (G_{N}(z)/F_{N}(z)). Let us for the moment restrict ourselves to the case Φ ≥ 3. It was argued in the paragraph between (4.13) and (4.15) that g(1/C) > 0. Furthermore, by the definition of f(z) in (4.12), we have f(1/C) = F_{N}(1/C) ≠ 0. Finally, by (4.17), we have f̃(1/C) < 0. If we use these observations, together with our assumption that Φ ≥ 3, from (4.19) we obtain the singular expansion

q_{N}(z) = z exp (\frac{G_{N}(z)}{F_{N}(z)})
= Cz exp (f̃(z)/f(z))(1 + α_{1}(1 − Cz)^{Φ/2−1}L(z) + O((1 − Cz)^{Φ/2}L(z))) (4.20)

as z → 1/C. Here,

α_{1} = − exp (\frac{f̃(1/C)}{f(1/C)})\frac{g(1/C)f̃(1/C)}{f^{2}(1/C)} = q_{N}(1/C)\frac{g(1/C)(F_{N}(1/C) log(C) − G_{N}(1/C))}{F_{N}^{2}(1/C)},

which is positive because of (4.18) and g(1/C) > 0. The singular expansions (4.8) and (4.9) now follow routinely: the claim on the sign of q_{d+ \frac{1}{2}} , respectively of q_{d+}, is a direct consequence of α_{1} being positive, while the claim on the sign of q_{1} follows from the fact that q_{N}(z) is monotone increasing on the interval [0, 1/C] (which is implied by Theorem 1.1).

Finally, if Φ = 2, then, leaving the details to the reader, the singular expansion in this case is

q_{N}(z) = 1 + \frac{α_{2}}{log(1 − Cz)} + O (\frac{1}{log^{2}(1 − Cz)})  (4.21)

for z → 1/C. Here, α_{2} = −f̃(1/C)/g(1/C), which is positive by our earlier observations. □

The remaining lemmas in this section are general results that describe the singular expansion of a function z(q) at q = q_{0}, given the singular expansion of its compositional inverse q(z) at the corresponding point. They are tailor-made for obtaining singular expansions of z_{N}(q) at q = − exp (−π cot(π/M)), respectively at q = q_{N}(1/C), by combining the appropriate lemma with Lemma 4.1, respectively with Lemma 4.2.

We start with the results which, upon combination with Lemma 4.2, imply singular expansions of z_{N}(q) at q = q_{N}(1/C). In the statements, we make use of right and left slit neighbourhoods, notions that have been defined in the paragraph above Proposition 1.6 in the Introduction.

Lemma 4.3. Suppose we are given a complex function q(z) which is analytic in a right slit neighbourhood of z = z_{0} and has a singular expansion that begins

q(z) = q_{0} + q_{1}(z − z_{0}) + q_{2}(z − z_{0})^{2} + ⋯

+ q_{d}(z − z_{0})^{d} + q_{d+ \frac{1}{2}} (z_{0} − z)^{d+ \frac{1}{2}} + O((z − z_{0})^{d+1}), (4.22)

where d ≥ 0, q_{d+ \frac{1}{2}} ≠ 0 and, if d > 0, then also q_{1} ≠ 0. Then there exists a local inverse function z(q) which, in a right slit neighbourhood of q = q_{0}, is analytic and has a singular expansion that begins

z(q) = z_{0} + z_{2}(q − q_{0})^{2} + O((q − q_{0})^{4})

if d = 0, where z_{2} = −1/q_{1/2}^{2}, and begins

z(q) = z_{0} + z_{1}(q − q_{0}) + z_{2}(q − q_{0})^{2} + ⋯

+ z_{d}(q − q_{0})^{d} + z_{d+ \frac{1}{2}} (q_{0} − q)^{d+ \frac{1}{2}} + O((q − q_{0})^{d+1})

if d > 0, where z_{1} = 1/q_{1} and z_{d+ \frac{1}{2}} = −q_{1}^{−d− \frac{3}{2}} q_{d+ \frac{1}{2}} .

Proof. We concentrate on the case where d > 0. By standard bootstrap arguments (see [12, Sections 2.5–2.7]), one sees that, in a right slit neighbourhood of q_{0},

z(q) = z_{0} + z_{1}(q − q_{0}) + z_{2}(q − q_{0})^{2} + ⋯ + z_{d}(q − q_{0})^{d} + z̃(q),  (4.23)

where the coefficients z_{0}, z_{1}, …, z_{d} agree with the corresponding Taylor coefficients of the compositional inverse of the truncated series

q_{0} + q_{1}(z − z_{0}) + q_{2}(z − z_{0})^{2} + ⋯ + q_{d}(z − z_{0})^{d},

and where z̃(q) is a function which is analytic in the same slit neighbourhood, and which satisfies z̃(q) = o((q−q_{0})^{d}). In particular, z_{1} = 1/q_{1}. Continuing the bootstrap, the expansion (4.23) (with q replaced by x in order to minimise the number of possible confusions) is now substituted in (4.22). In that manner, we obtain

0 = q_{1}z̃(x) + q_{d+ \frac{1}{2}} z_{1}^{d+ \frac{1}{2}} (q_{0} − x)^{d+ \frac{1}{2}} + O((x − q_{0})^{d+1}).

The claimed result is now obvious. □

Lemma 4.4. Suppose we are given a complex function q(z) which is analytic in a right slit neighbourhood of z = z_{0}, where it has a singular expansion that begins

q(z) = q_{0} + q_{1}(z − z_{0}) + q_{2}(z − z_{0})^{2} + ⋯

+ q_{d−1}(z − z_{0})^{d−1} + q_{d+}(z − z_{0})^{d} log(z_{0} − z) + O((z − z_{0})^{d}), (4.24)

where d ≥ 1, q_{d+} ≠ 0, and, if d > 1, then also q_{1} ≠ 0. Then there exists a local inverse function z(q) which, in a right slit neighbourhood of q = q_{0}, is analytic and has a singular expansion that begins

z(q) = z_{0} + z_{1+}(q − q_{0}) log^{−1}(q_{0} − q) + o((q − q_{0}) log^{−1}(q_{0} − q))  (4.25)

if d = 1, where z_{1+} = 1/q_{1+}, and begins

z(q) = z_{0} + z_{1}(q − q_{0}) + z_{2}(q − q_{0})^{2} + ⋯

+ z_{d−1}(q − q_{0})^{d−1} + z_{d+}(q − q_{0})^{d} log(q_{0} − q) + O((q − q_{0})^{d}) (4.26)

if d > 1, where z_{1} = 1/q_{1} and z_{d+} = −q_{1}^{−d−1}q_{d+}.

Proof. For (4.26), one proceeds exactly in the same fashion as in the proof of Lemma 4.3.

In order to establish the expansion (4.25), one replaces z by z(x) in (4.24), and subsequently one applies the logarithm on both sides. This leads to

log(q_{0} − x) = log(q_{1+}) + log(z_{0} − z(x)) + log log(z_{0} − z(x)) + O( log^{−1}(z_{0} − z(x))).

In order to simplify, we replace q_{0} − x by X and log(z_{0} − z(x)) by Z(X):

log(X) = log(q_{1+}) + Z(X) + log(Z(X)) + O(Z(X)^{−1}).  (4.27)

Applying bootstrapping again in a neighbourhood of X = 0, we must have Z(X) = log(X) + Z̃(X), where Z̃(X) = o(log(X)). If we substitute this in (4.27), we obtain, after little simplification,

0 = log(q_{1+}) + Z̃(X) + log log(X) + o(1).  (4.28)

Now we see that Z̃(X) = − log log(X) + ˜ Z̃(X), where ˜ Z̃(X) = o(log log(X)). By substituting this in (4.28), we arrive at

0 = log(q_{1+}) + ˜ Z̃(X) + o(1),  (4.29)

from which we deduce ˜ Z̃(X) = − log(q_{1+}) + o(1). If we now put everything together, then we obtain

Z(X) = log(X) − log log(X) − log(q_{1+}) + o(1),

or, in the original notation,

z_{0} − z(x) = \frac{q_{0} − x}{q_{1+} log(q_{0} − x)}(1 + o(1)).

After replacement of x by q, one sees that this is equivalent to (4.25). □

Lemma 4.5. Suppose we are given a complex function q(z) which is analytic in a right slit neighbourhood of z = z_{0}, where it has a singular expansion that begins

q(z) = q_{0} + q_{1}log^{−1}(z_{0} − z) + O(log^{−2}(z_{0} − z)),

where q_{1} ≠ 0. Then there exists a local inverse function z(q) which, in a right slit neighbourhood of q = q_{0}, is analytic and has a singular expansion of the form

z(q) = exp (\frac{q_{1}}{q − q_{0}} + O(1)).

In particular, z(q) has an essential singularity at q = q_{0}.

Proof. This is again easily derived by bootstrapping. □

Finally we provide a general result which, upon combination with Lemma 4.1, yields the singular expansion of z_{N}(q) at q = − exp ( − π cot(π/M)).

Lemma 4.6. Let q(z) be a complex function that is analytic near ∞ except for a cut on the positive real axis.

(i) Suppose that q(z) has a singular expansion at z = ∞ that begins

q(z) = q_{0} + q_{1}(−z)^{− \frac{r}{n}} + O((−z)^{− \frac{r+1}{n}} log^{b}(−z)),

where q_{1} ≠ 0 and r, n, b are non-negative integers such that r ≥ 1. Then there exists a local inverse function z(q) which, in a neighbourhood of q = q_{0}, has a singular expansion that begins

z(q) = −q1^{\frac{n}{r}} (q − q_{0})^{− \frac{n}{r}} + O((q − q_{0})^{− \frac{n−1}{r}} log^{b} (q − q_{0})).

(ii) Suppose that q(z) has a singular expansion at z = ∞ that begins

q(z) = q_{0} + q_{1}(−z)^{− \frac{r}{n}} log^{b}(−z) + O((−z)^{− \frac{r}{n}} log^{b−1}(−z)),

where q_{1} ≠ 0 and r, n, b are non-negative integers with r, b ≥ 1. Then there exists a local inverse function z(q) which, in a neighbourhood of q = q_{0}, has a singular expansion that begins

z(q) = − (−\frac{n}{r})^{\frac{bn}{r}} q1^{\frac{n}{r}} (q − q_{0})^{− \frac{n}{r}} log^{\frac{bn}{r}} (q − q_{0}) + O((q − q_{0})^{− \frac{n}{r}} log^{\frac{bn}{r} −1} (q − q_{0})).

(iii) Suppose that q(z) has a singular expansion at z = ∞ that begins

q(z) = q_{0} + q_{1} log^{−1}(−z) + O( log^{−2}(−z)),

where q_{1} ≠ 0. Then there exists a local inverse function z(q) which, in a neighbourhood of q = q_{0}, has a singular expansion that begins

z(q) = exp (\frac{q_{1}}{q − q_{0}} + O(1)).

Remark. If, in Case (i) with r = 1, the series q(z) admits in fact a Puiseux expansion in (−z)^{− \frac{1}{n}}, then it is not difficult to see that z(q) has a pole of order n at q = q_{0}. If the singular expansion of q(z) at z = −∞ should have terms containing log(−z) (which is often the case in the situation of Lemma 4.1; see the remark accompanying that lemma), then the point q = q_{0} will be a branch point of z(q).

Proof. Once more, this is easily derived by bootstrapping. □

5. Proof of Theorem 1.2

For the proof of the theorem, we shall require the following auxiliary result.

Lemma 5.1. The function F_{N}(z) does not vanish on its disk of convergence |z| < 1/C_{N}.

Proof. The following argument is borrowed from [22, page 94, last corollary]. By Corollary 2.4, the Taylor coefficients B̂_{N}(m) of F̂_{N}(z) are positive and satisfy Equation (2.1), i.e.,

δ_{n,0} + ∑_{k=1}^{n} B̂_{N}(k)B_{N}(n − k) = B_{N}(n).

Since the coefficients B_{N}(m) are also positive and B_{N}(0) = 1, it follows that 0 ≤ B̂_{N}(n) ≤ B_{N}(n) for all n ≥ 0. Hence, the radius of convergence of the Taylor series of 1/F_{N}(z) at z = 0 is at least as large as the radius of convergence of F_{N}(z) at z = 0. It is necessarily equal to the latter because z = 1/C is a branch point of F_{N}(z), and thus also of 1/F_{N}(z).

In particular, F_{N}(z) cannot vanish at some point z_{0} such that |z_{0}| < 1/C, because otherwise the radius of convergence of 1/F_{N}(z) would be at most |z_{0}|, a contradiction. □

Remark. In the proof of Theorem 1.3.(i) (given in Section 6), we show that a classical result of Pólya [29] on hypergeometric series implies that F_{N}(z) vanishes nowhere in the slit plane ℂ \ [1/C, +∞). Nevertheless, we believe that the above argument, proving a weaker result, is still worth being recorded since it is only based on the positivity of the coefficients B_{N}(m) and B̂_{N}(m) and not on the hypergeometric nature of F_{N}(z).

We now turn to the proof of items (i)–(vi) of Theorem 1.2.

(i), (ii) In view of the explicit expression (1.5), the claim is trivial for Φ = 1. We therefore assume Φ ≥ 2 from now on.

Clearly, the discussion in the proof of Lemma 5.1 also implies that the radius of convergence of the Taylor series at z = 0 of exp(G_{N}(z)/F_{N}(z)) is at least 1/C. By Lemma 4.2, which says in particular that q_{N}(z) has a singularity at z = 1/C, it cannot be larger.

It remains to prove that the Taylor series ∑_{m≥1} q_{m}z^{m} of q_{N}(z) converges on the circle |z| = 1/C. By (4.17), we have

lim_{z→1/C} \frac{G_{N}(z)}{F_{N}(z)} = log(C) + lim_{z→1/C} \frac{G_{N}(z) − log(C)F_{N}(z)}{F_{N}(z)} = log(C) + \frac{S}{F_{N}(1/C)}, (5.1)

where the second term must be understood as 0 if lim_{z→1/C} F_{N}(z) = +∞. Hence q_{N}(1/C) exists and is finite. By Abel’s theorem, q_{N}(1/C) = ∑_{m≥1} q_{m}/C^{m}, and since the q_{m} are non-negative, ∑_{m≥1} q_{m}z^{m} converges for any z such that |z| = 1/C.

(iii) We have

max_{|z|=1/C} |q_{N}(z)| = q_{N}(1/C).

By (5.1), we have

q_{N}(1/C) = exp (\frac{S}{F_{N}(1/C)}) ≤ 1

because S < 0 and F_{N}(1/C) > 0. There is equality to 1 only if F_{N}(1/C) = +∞, which, by (4.14), happens exactly when Φ = 1 or Φ = 2.

(iv) The claimed assertions follow from (1.5) upon little calculation.

(v) By the standard theorems of singularity analysis (see [14, Ch. VI]), the assertion follows immediately from Lemma 4.2.(i).

(vi) Again, by the standard theorems of singularity analysis, the assertion follows immediately from Lemma 4.2.(ii), (iii). □

6. Proof of Theorem 1.3

(i) From Section 3, we know that F_{N}(z) and G_{N}(z) can both be analytically continued to ℂ \ [1/C, +∞). Hence

q_{N}(z) = exp (\frac{G_{N}(z) + log(z)F_{N}(z)}{F_{N}(z)}) = z exp (\frac{G_{N}(z)}{F_{N}(z)})

can be continued at least to ℂ \ ([1/C, +∞) ∪ Z), where Z is the set of zeroes of F_{N}(z).

The reader should recall that Lemma 5.1 says that the intersection of Z and the open disk of convergence of F_{N}(z) is empty. We now show that, in fact, the entire set Z is empty. For this, we apply a result of Pólya [29, p. 192] on hypergeometric functions. Recalling the hypergeometric notation

q+1F_{q}[\begin{matrix}a_{0}, a_{1}, …, a_{q} \\ b_{1}, …, b_{q} \end{matrix}; z] = ∑_{k=0}^{∞} \frac{(a_{0})_{k} (a_{1})_{k} ⋯ (a_{q})_{k}}{k! (b_{1})_{k} ⋯ (b_{q})_{k}} z^{k},

Pólya’s result implies in particular that the above hypergeometric function does not vanish for any z ∈ ℂ \ [1, +∞) when 0 < a_{0} < 1, 0 < a_{1} < b_{1}, …, 0 < a_{q} < b_{q}. Now, indeed, F_{N}(z) can be written in hypergeometric notation:

F_{N}(z) = _{Φ}F_{Φ−1}[\begin{matrix}r_{1,1}/N_{1}, …, r_{k,φ(N_{k})}/N_{k} \\ 1, …, 1 \end{matrix}; Cz] .

In particular, we see that Pólya’s conditions are satisfied by this hypergeometric function, which proves that Z is empty.

(ii) If Φ ≥ 2, this is a consequence of the singular expansion of q_{N}(z) at z = 1/C given in Lemma 4.2.

If Φ = 1, then we know that q_{(2)}(z) = (1−\sqrt{1 − 4z})^{2}/(4z), which has evidently a branch point with non-trivial monodromy at z = 1/C_{(2)} = 1/4.

(iii) The assertion concerning the limit of q_{N}(z) at infinity is an immediate consequence of Lemma 3.1. □

7. Proof of Theorem 1.4

(i) We know from Theorem 1.3 that

lim_{z→∞} q_{N}(z) = − exp(−π cot(π/M)) =: ρ,

where the limit has to be performed along a path that avoids the cut [1/C, +∞). Let us suppose that the radius of convergence of z_{N}(q), R say, is strictly larger than |ρ|. One can find ε > 0 and A(ε) > 0 with the property that, if |x| > A(ε) and x ∉ [1/C, +∞), then

|q_{N}(x) − ρ| < ε and |ρ| + ε < R.

For the above x, the quantity z_{N}(q_{N}(x)) is well-defined and equals x. Consequently,

∞ = lim_{x→∞}_{|x|>A(ε)}_{x∉[1/C,+∞)}x = lim_{x→∞}_{|x|>A(ε)}_{x∉[1/C,+∞)}z_{N}(q_{N}(x)) = lim_{q→ρ} z_{N}(q),

where the last limit is along a suitable path. Hence, the point q = ρ is a singularity of z_{N}(q), which contradicts our assumption that R > |ρ|. Therefore the radius of convergence R is in fact ≤ |ρ|. □

(ii) Arguing by contradiction, we suppose that the radius of convergence of z_{N}(q) is strictly larger than q_{N}(1/C). In that case, z_{N}(q) is analytic around q_{N}(1/C). For the derivative of z_{N}, we have

z′_{N}(q_{N}(z)) = \frac{1}{q′_{N}(z)}.  (7.1)

Let us first assume that Φ > 4. From Theorem 1.2.(vi), we know that the m-th coefficient of q_{N}(z), q_{m} say, behaves like C^{m}/m^{Φ/2} (up to a multiplicative constant). Hence, since Φ/2 > 2, the series ∑_{m=0}^{∞} mq_{m}/C^{m−1} converges, and by Abel’s theorem it agrees with the limit

ω_{1} := lim_{z→1/C}_{|z|<1/C}q′_{N}(z).

Hence, by (7.1) and the continuity of z′_{N}(q), we have z′_{N}(q_{N}(1/C)) = 1/ω_{1}, which is different from zero. As a consequence, by [17, Theorems 2.4b, 2.4c], z_{N}(q) can be inverted in a neighbourhood of q = q_{N}(1/C). This would imply that q_{N}(z) is analytic around 1/C, which contradicts Theorem 1.2.(ii).

On the other hand, if Φ = 4, the above argument has to be adapted in order to lead to a contradiction. To begin with, by Lemma 4.2.(iii), we have

q_{N}(z) − q_{N}(1/C) = ω_{2}(1 − Cz) log(1 − Cz)(1 + o(1))  (7.2)

for z → 1/C, where ω_{2} is some non-zero constant. Next, we compute the derivative of q_{N}(z) using the expression given in (4.19) for the quotient G_{N}(z)/F_{N}(z). Subsequently, we compute its singular expansion for z → 1/C in the same style as the one for q_{N}(z). The result is that

q′_{N}(z) = ω_{3} log(1 − Cz)(1 + o(1))  (7.3)

for z → 1/C, where ω_{3} is some non-zero constant. Consequently,

z′_{N}(q_{N}(1/C)) = lim_{z→1/C}_{|z|<1/C} \frac{1}{q′_{N}(z)} = 0.

Since z′_{N}(q) is analytic in a neighbourhood of q = q_{N}(1/C), and since z′_{N}(q) cannot be constant (this would imply that q_{N}(z) is linear, which contradicts Theorem 1.2.(ii)), we have

z′_{N}(q) = ω_{4}(q − q_{N}(1/C))^{s}(1 + o(1))

for q → q_{N}(1/C), where ω_{4} is a non-zero constant and s is a positive integer. If we use this in (7.1), together with (7.2) and (7.3), we obtain

ω_{4}ω_{2}^{s}(1 − Cz)^{s} log^{s}(1 − Cz)(1 + o(1)) = \frac{1 + o(1)}{ω_{3} log(1 − Cz)}

for z → 1/C, which is absurd. □

Remark. The above argument for the case where Φ = 4 does not lead to a contradiction when applied to the case where Φ = 3. This is in accordance with Theorem 1.5 and the fact that exp ( − π cot(π/M)) is larger than q_{N}(1/C_{N}) in the relevant cases, see Section 9.

8. Comparison of the two critical values exp(−π cot(π/M_{N})) and q_{N}(1/C_{N})

Because of (1.7), the point q = exp(−π cot(π/M_{N})) is a potential singularity of z_{N}(q). In Theorem 1.4.(ii) we have shown that the radius of convergence of z_{N}(q) is at most q_{N}(1/C_{N}). It is therefore important to know whether exp(−π cot(π/M_{N})) is less than q_{N}(1/C_{N}) or not. In this section, we show that for Φ_{N} ≥ 4 we have in fact exp(−π cot(π/M_{N})) > q_{N}(1/C_{N}), which fits well with Conjecture 1.7.

Lemma 8.1. Let N_{1}, N_{2}, …, N_{k} be positive integers, all of which at least 2, such that Φ = Φ_{N} ≥ 4. Furthermore, let M = M_{N}, as before. Then

exp(−π cot(π/M)) > q_{N}(1/C).  (8.1)

Proof. From [31, inequality on top of p. 157, which, as the proof shows, remains valid for real n] we know that for x ≥ 1 we have

Γ(x + 1) = \sqrt{2πx} (\frac{x}{e})^{x} e^{λ_{x}}, with \frac{1}{12x + 1} < λ_{x} < \frac{1}{12x}.  (8.2)

We use these effective bounds on the gamma function to provide an upper bound for B_{N}(m). Let us first suppose that m ≥ 2. Then we have

B_{N}(m) = ∏_{j=1}^{k} B_{N_{j}}(m) = C^{m} ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{(r_{i,j}/N_{j})_{m}}{m!} = C^{m} ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{Γ (m + \frac{r_{i,j}}{N_{j}} )}{Γ (\frac{r_{i,j}}{N_{j}} ) Γ(m + 1)}

< C^{m}e^{Λ_{m}} ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{1}{Γ(r_{i,j}/N_{j})} (\frac{m + \frac{r_{i,j}}{N_{j}} − 1}{m})^{m+ \frac{1}{2}} (\frac{m + \frac{r_{i,j}}{N_{j}} − 1}{e})^{\frac{r_{i,j}}{N_{j}} −1} , (8.3)

where

Λ_{m} = ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (\frac{1}{12(m + \frac{r_{i,j}}{N_{j}} − 1)} − \frac{1}{12m + 1}) .

The quantity Λ_{m} may be estimated from above as follows:

Λ_{m} = ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (\frac{13 − 12\frac{r_{i,j}}{N_{j}}}{12(m + \frac{r_{i,j}}{N_{j}} − 1)(12m + 1)}) ≤ ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (\frac{13 − 12\frac{r_{i,j}}{N_{j}}}{12(m − 1)(12m + 1)})

≤ ∑_{j=1}^{k} φ(N_{j}) (\frac{13 − 12\frac{1}{2}}{12(m − 1)(12m + 1)}) ≤ \frac{7Φ}{12(m − 1)(12m + 1)} ≤ \frac{7}{300}Φ.  (8.4)

On the other hand, using the well-known elementary inequality (1 + \frac{x}{m})^{m} ≤ e^{x}, valid for x > −m, we have

(\frac{m + \frac{r_{i,j}}{N_{j}} − 1}{m})^{m+ \frac{1}{2}} (\frac{m + \frac{r_{i,j}}{N_{j}} − 1}{e})^{\frac{r_{i,j}}{N_{j}} −1} = (1 + \frac{\frac{r_{i,j}}{N_{j}} − 1}{m})^{m} \frac{e^{1− \frac{r_{i,j}}{N_{j}}} (1 + ^{\frac{\frac{r_{i,j}}{N_{j}} −1}{m}})^{\frac{1}{2}}}{(m + \frac{r_{i,j}}{N_{j}} − 1)^{1− \frac{r_{i,j}}{N_{j}}}}

≤ \frac{1}{(m − 1)^{1− \frac{r_{i,j}}{N_{j}}}} .  (8.5)

Finally, by the reflection formula (cf. [3, Theorem 1.2.1])

Γ(x)Γ(1 − x) = \frac{π}{sin(πx)}

for the gamma function, in the case that all N_{j}’s are different from 2 (in which case φ(N_{j}) is always even) we get

∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{1}{Γ(r_{i,j}/N_{j})} = ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})/2} \frac{sin (πr_{i,j}/N_{j})}{π} ≤ π^{−Φ/2}.  (8.6)

One can then see that the inequality above holds even if some of the N_{j}’s should happen to equal 2. If (8.4)–(8.6) are substituted back in (8.3), then the result is

B_{N}(m) < C^{m}e^{\frac{7}{300} Φ}π^{−Φ/2}(m − 1)^{− ∑_{j=1}^{k} ∑_{i=1}^{φ(N_{j})} (1− \frac{r_{i,j}}{N_{j}} )}
< C^{m}e^{\frac{7}{300} Φ}π^{−Φ/2} (m − 1)^{−Φ/2} ,  (8.7)

where we used (4.11) to obtain the last line.

The case of m = 1 has to be treated separately. In that case, we see that

B_{N}(1) = ∏_{j=1}^{k} B_{N_{j}}(1) = C ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} \frac{r_{i,j}}{N_{j}} = C ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})/2} \frac{r_{i,j}(N − r_{i,j})}{N_{j}^{2}} ≤ C · 2^{−Φ}  (8.8)

if all N_{j}’s are different from 2. Again, it is readily seen that the inequality also holds if some of the N_{j}’s should happen to equal 2.

We now combine (8.7) and (8.8) in order to estimate F_{N}(1/C):

F_{N}(1/C) = ∑_{m=0}^{∞} B_{N}(m)C^{−m}

< 1 + \frac{1}{2^{Φ}} + (\frac{e^{\frac{7}{300}}}{π^{1/2}} )^{Φ} ∑_{m=2}^{∞} (m − 1)^{−Φ/2}

< 1 + \frac{1}{2^{Φ}} + (\frac{e^{\frac{7}{300}}}{π^{1/2}} )^{Φ} ζ(Φ/2)

< 1 + \frac{1}{2^{max\{4,k\}}} + (\frac{e^{\frac{7}{300}}}{π^{1/2}} )^{max\{4,k\}} ζ( max\{4, k\}/2).  (8.9)

In the sequel, we denote the quantity on the right-hand side of (8.9) by c_{k}.

We are now in the position to establish the inequality (8.1) for “large enough” N. Namely, since

π cot(π/M) ≤ M

and (see (4.18))

G_{N}(1/C) − log(C)F_{N}(1/C) < − log(C),

it suffices to prove the inequality

exp(−M) ≥ exp (−\frac{log C}{F_{N}(1/C)}) ,

or, equivalently,

log C ≥ M · F_{N}(1/C).

We now make use of the estimation (cf. [5, Theorem 8.8.7])

φ(n) ≥ max \{1, \frac{n}{e^{γ} log log n + \frac{3}{log log n}} \}  (8.10)

for the totient function, where γ denotes Euler’s constant. For convenience, let us write \overline{φ}(n) for the right-hand side of (8.10). Then, use of (8.10) in the definition of C gives

log C ≥ ∑_{j=1}^{k} log N_{j}^{φ(N_{j})} = ∑_{j=1}^{k} φ(N_{j}) log N_{j} ≥ ∑_{j=1}^{k} \overline{φ}(N_{j}) log N_{j}.

If we put this together with (8.9), we see that the inequality (8.1) will be proved whenever

∑_{j=1}^{k} \overline{φ}(N_{j}) log N_{j} ≥ c_{k}M.  (8.11)

It remains to consider the cases where (8.11) does not hold. We claim that this is only a finite number of cases. Indeed, for fixed k, there can only be a finite number of k-tuples (N_{1}, N_{2}, …, N_{k}) for which (8.11) is violated, since trivially c_{k} is constant for fixed k, and since \overline{φ}(M) log M grows faster than M. On the other hand, if k ≥ 15, then (8.11) holds automatically. For, we have

min_{M≥2}\{\overline{φ}(M) log M − c_{15}M\} ≥ −9,

and therefore (assuming that M = N_{1} without loss of generality)

∑_{j=1}^{k} \overline{φ}(N_{j}) log N_{j} = \overline{φ}(M) log M + ∑_{j=2}^{k} \overline{φ}(N_{j}) log N_{j}
≥ c_{15}M − 9 + (k − 1) log 2
≥ c_{k}M − 9 + (k − 1) log 2
≥ c_{k}M

for k ≥ 15, where we used the simple fact that the c_{k}’s are monotone decreasing in k in the third line.

In summary, there is indeed only a finite number of cases left where (8.11) does not hold. For these cases, we have verified on the computer that the inequality (8.1) is satisfied. For carrying out this verification, we used the estimation

q_{N}(1/C) = exp (\frac{∑_{m=0}^{∞} B_{N}(m)(H_{N}(m) − log C)C^{−m}}{∑_{m=0}^{∞} B_{N}(m)C^{−m}} )

< exp (\frac{∑_{m=0}^{20} B_{N}(m)(H_{N}(m) − log C)C^{−m}}{∑_{m=0}^{20} B_{N}(m)C^{−m} + e^{\frac{7}{300} Φ}π^{−Φ/2} ∑_{m=21}^{∞} (m − 1)^{−Φ/2}} )

< exp (\frac{∑_{m=0}^{20} B_{N}(m)(H_{N}(m) − log C)C^{−m}}{∑_{m=0}^{20} B_{N}(m)C^{−m} + e^{\frac{7}{300} Φ}π^{−Φ/2} (\frac{Φ}{2} − 1)^{−1} 19^{− \frac{Φ}{2} +1}} )  (8.12)

(with the second line following again from (4.18) and (8.7), and the last line from ∑_{m=21}^{∞} (m − 1)^{−Φ/2} < ∫_{19}^{∞} x^{−Φ/2} dx), and we actually compared the left-hand side of (8.1) with the upper bound on the right-hand side given in (8.12). This completes the proof of the lemma. □

9. Proof of Theorem 1.5: the cases Φ_{N} = 1, 2, 3

The case of Φ = 1 is a trivial case. We provide the details nevertheless for the sake of completeness. For the cases Φ = 2, 3, it is well documented in the literature that the corresponding functions z_{N}(q) live in the world of modular forms. We refer the reader to [34, 40] for reviews of the corresponding classical theory. Despite of this, it seems that the questions that we treat in the present paper — namely questions concerning the analytic nature of the function z_{N}(q) — have not been recorded in sufficient detail to extract complete information about the singularities of the function, for example, and, thus, of the radius of convergence of the Taylor series at q = 0.

The purpose of this section is to compile the relevant facts, to provide a coherent overview, and to fill possible gaps whenever necessary.

Since we shall make use of it below, we record here some basic facts on modular forms. Given a subgroup Γ of the modular group SL_{2}(ℤ) and a non-negative integer k, a function f(τ) from the complex upper half plane ℍ to the complex numbers is called a modular form of weight k for Γ if it is meromorphic on ℍ and

f(T (τ)) = (cτ + d)^{k}f(τ)  (9.1)

for all elements T = ( \begin{matrix}a & b \\ c & d \end{matrix}) of Γ, where the action of T is defined by T (τ) = (aτ +b)/(cτ +d). Modular forms of weight k for Γ satisfy the valence formula (cf. [30, Theorem 4.1.4])

∑_{ζ∈𝔽} \frac{ord(f, ζ)}{| stab_{ζ}(Γ̂)|} = \frac{|{ \atop SL_{2}(ℤ)}/Γ̂|}{12}k.  (9.2)

Here,

− 𝔽 is a fundamental region for the action of Γ on ℍ to which one adds the cusps (a system of representatives of the orbit of ∞ under the action of SL_{2}(ℤ) when restricted to Γ);

− Γ̂ is the mapping group corresponding to Γ, i.e., the group arising from Γ upon identification of T and −T if both of them should be contained in Γ (the action (9.1) on ℍ of T and −T is identical);

− ord(f, ζ) is the usual order of f at ζ if ζ is not a cusp;

− if ζ is a cusp, ζ = T ∞ for T = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ), then ord(f, ζ) is defined as the order of the series expansion of (cτ + d)^{−k}f(T (τ)) in q̃ = exp(2iπτ/n_{T}) (as a Laurent series in q̃), where k is the weight of f, and where n_{T} is the least positive integer such that ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )^{n_{T}} ∈ T ΓT^{−1};

− stab_{ζ}(Γ̂) is the subgroup of Γ̂ consisting of the elements fixing ζ.

The stabiliser stab_{i} Γ̂ of i in Γ̂ can be 2 or 1 depending on whether or not V = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) or −V is in Γ. The stabiliser stab_{ρ} Γ̂ of ρ = exp(2iπ/3) in Γ̂ can be 3 or 1 depending on whether or not P = ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ) or −P is in Γ. All other stabilisers consist only of one element.

Below we shall make frequent use of the following special functions. The Eisenstein series (in Ramanujan’s notation) Q(q) and R(q) are defined by

Q(q) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n} and R(q) = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}

and σ_{k}(n) = ∑_{d|n} d^{k}. The Eisenstein series E_{4}(τ) := Q( exp(2iπτ)) is a modular form of weight 4 for the full modular group SL_{2}(ℤ), while E_{6}(τ) := R( exp(2iπτ)) is a modular form of weight 6 for the same group. It is well-known (see [32, p. 143]) that the zeroes of Q(q) are of the form q = exp(2iπτ), where τ runs through the elements of the orbit of ρ under SL_{2}(ℤ). All of these are simple zeroes. Similarly (see [32, p. 143]), R(q) has only simple zeroes which are of the form q = exp(2iπτ), where τ runs through the elements of the orbit of i under SL_{2}(ℤ). The unit circle is a natural boundary for both Q(q) and R(q).

The Dedekind–Klein j-invariant is given by

j(τ) = 1728\frac{E_{4}^{3}(τ)}{E_{4}^{3}(τ) − E_{6}^{2}(τ)}.

We shall rather use the variant

J(q) = \frac{Q^{3}(q)}{Q^{3}(q) − R^{2}(q)},  (9.3)

so that j(τ) = 1728J(exp(2iπτ)). The function j(τ) is a modular form of weight 0 for SL_{2}(ℤ). J(q) is meromorphic in the unit disk with a unique pole at q = 0, which is a simple pole, and the unit circle is a natural boundary.

The Dedekind η-function is defined by

η(τ) = exp(iπτ/12) ∏_{n=1}^{∞}(1 − exp(2iπnτ)).

We shall rather use the variant

H(q) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}).

The function η^{24}(τ) can be expressed in terms of Eisenstein series in the form

η^{24}(τ) = E_{4}^{3}(τ) − E_{6}^{2}(τ),

and it is therefore a modular form of weight 12 for SL_{2}(ℤ).

Two of Jacobi’s theta functions will sometimes appear: the function θ_{2}(q) defined by

θ_{2}(q) = ∑_{n=−∞}^{∞} q^{(n+ \frac{1}{2} )^{2}} = 2q^{1/4} ∏_{j=1}^{∞}(1 − q^{2j})(1 + q^{2j})^{2}

(the equality of the two expressions above follows from Jacobi’s triple product identity, see [3, Theorem 10.4.1]), and the function θ_{3}(q) defined by

θ_{3}(q) = ∑_{n=−∞}^{∞} q^{n^{2}} = ∏_{j=1}^{∞}(1 − q^{2j})(1 + q^{2j−1})^{2}

(with again Jacobi’s triple product identity explaining the equality).

We shall also frequently use the fact that, if f(τ) is a modular form of weight k for SL_{2}(ℤ), then f(Nτ) is a modular form of weight k for Γ_{0}(N), where Γ_{0}(N) is the subgroup of SL_{2}(ℤ) consisting of all matrices ( \begin{matrix}a & b \\ c & d \end{matrix}) with c ≡ 0 mod N. Furthermore, we have |{ \atop SL_{2}(ℤ)}/Γ̂_{0}(2)| = 3 and |{ \atop SL_{2}(ℤ)}/Γ̂_{0}(3)| = 4 (see [30, (1.4.23)]).

We are now ready to discuss all cases in which Φ = 1, 2, 3.

9.1. Φ = 1. There is only a single example in which Φ = 1, namely if N = (2). In that case (see (1.5)), we have q_{(2)}(z) = (1 − \sqrt{1 − 4z})^{2}/(4z) and z_{(2)}(q) = q/(1 + q)^{2}. Hence, q_{(2)}(z) has exactly one singularity at z = 1/4, which is of square root type, and the radius of convergence is 1/4 = 1/C. Moreover, z_{(2)}(q) = q/(1 + q)^{2} has exactly one singularity at q = −1. Its radius of convergence is 1 = exp(−π cot(π/2)).

9.2. Φ = 2. In this case, according to Theorem 1.2.(ii) and the remark after the statement of the theorem, the point z = 1/C is always a singularity of q_{N}(z), with q_{N}(1/C) = 1. Moreover, because of (4.7), this singularity is of “1/ log”-type. Consequently, by Lemma 4.5, the mirror map z_{N}(q) has an essential singularity at q = q_{N}(1/C) = 1.

Case N = (3). Here, we have

F_{(3)}(z) = ∑_{n=0}^{∞} \frac{(3n)!}{n!^{3}}z^{n} = _{2}F_{1}[\begin{matrix}1/3, 2/3 \\ 1 \end{matrix}; 27z] .

It is well-known (cf. e.g. [40, (23)]) that

z_{(3)}(q) = (Q_{3}^{3/2}(q) − R_{3}(q))/(54Q_{3}^{3/2}(q)),  (9.4)

where Q_{3}(q) = (Q(q) + 9Q(q^{3}))/10 and R_{3}(q) = (R(q) + 27R(q^{3}))/28. An alternative expression (cf. [9, Theorem 2.3]) is given by

z_{(3)}(q) = \frac{(θ_{2}(q^{1/3})θ_{2}(q) + θ_{3}(q^{1/3})θ_{3}(q) − θ_{2}(q)θ_{2}(q^{3}) − θ_{3}(q)θ_{3}(q^{3}))^{3}}{216 (θ_{2}(q)θ_{2}(q^{3}) + θ_{3}(q)θ_{3}(q^{3}))^{3}}

= \frac{q}{27q + ∏_{n=1}^{∞} \frac{(1 − q^{n})^{12}}{(1 − q^{3n})^{12}}} = \frac{1}{27 + \frac{H^{12}(q)}{H^{12}(q^{3})}} .

The second equality can be easily derived using standard reasoning that involves an estimation of the orders of both expressions at the cusps of a fundamental region of Γ_{0}(12), application of the valence formula (9.2), and a verification that sufficiently many coefficients in the power series expansion in q agree; see [15, Sections 3 and 4] for a detailed description of this kind of argument.

Again alternatively, z_{(3)}(q) is solution of the equation (cf. [24, Sec. 5.1, p. 176])

1728J(q) = \frac{(1 + 216z_{(3)}(q))^{3}}{z_{(3)}(q)(1 − 27z_{(3)}(q))^{3}} .  (9.5)

By Lemma 4.1 (with M = 3, L = 3, R = 2/3) and Lemma 4.6, we know that z_{(3)}(q) has a pole of order 3 at q = − exp(−π/\sqrt{3}). (This corrects the statement in the paragraph containing (3) in [38].) Hence, by (9.4), q = − exp(−π/\sqrt{3}) must be a zero of Q_{3}(q) of order at least 2. Since the function E_{4}(τ) is a modular form of weight 4 for SL_{2}(ℤ), the function Q̃_{3}(τ) := Q_{3}( exp(2iπτ)) is a modular form of weight 4 for Γ_{0}(3). It is easy to see that the orbit of ρ under SL_{2}(ℤ) splits under the action of Γ_{0}(3) into three orbits: one orbit containing ρ, another orbit containing ρ + 1 = \frac{1}{2} + \frac{i\sqrt{3}}{2}, and another orbit containing ρ_{1} = \frac{1}{2} + \frac{i}{2\sqrt{3}}. The reader should note that exp(2iπρ_{1}) = − exp(−π/\sqrt{3}). If we now apply

the valence formula (9.2) to Q̃_{3}(τ), then, since | stab_{ρ_{1}}(Γ̂_{0}(3))| = 3, while | stab_{ρ}(Γ̂_{0}(3))| = 1 and | stab_{i}(Γ̂_{0}(3))| = 1, we obtain

\frac{1}{3} ord (Q̃_{3}, ρ_{1}) + ∑_{ζ∈𝔽\\{ρ_{1}\}} ord(Q̃_{3}, ζ) = \frac{4}{3}.

We know that Q̃_{3}(τ) is analytic in the upper half plane, that ord(Q̃_{3}, ζ) = 0 for all cusps ζ, and that ord(Q̃_{3}, ρ_{1}) ≥ 2. Therefore, if the above equation wants to hold then necessarily ord(Q̃_{3}, ρ_{1}) = 4, and all other orders must be zero. (As a by-product, we see that R_{3}(q) has a zero of order 3 at q = − exp(−π/\sqrt{3}). We remark that the valence formula applied to R_{3}(q) implies that there exists another, simple zero in a fundamental region of Γ̂_{0}(3). We claim that this is q = exp(−2π\sqrt{3}) and all elements in its orbit. To see this, one first observes that (9.3) and the previously quoted fact that Q( − exp(−π\sqrt{3})) = 0 imply that J( − exp(−π\sqrt{3})) = 0. Combining this with the modular equation of level 2, which provides a polynomial relation between J(q) and J(q^{2}) (cf. [18, expression for H_{2}(x, y) given on p. 192]), we infer that J( exp(−2π\sqrt{3})) = 125/4. If this is substituted in (9.5), one obtains four possible values for z_{(3)}( exp(−2π\sqrt{3})), one of which is 1/54. By a numerical calculation, the other three values can be ruled out. Finally, by inserting z_{(3)}( exp(−2π\sqrt{3})) = 1/54 in (9.4), we see that R( exp(−2π\sqrt{3})) = 0, as we claimed.)

In summary, the above arguments prove that z_{(3)}(q) is meromorphic on the unit disk |q| < 1 with poles of third order at all points exp(2iπτ), where τ is an element of the orbit of ρ_{1} under Γ_{0}(3). It is not difficult to see that exp(2iπρ_{1}) = − exp(−π/\sqrt{3}) is the element of smallest modulus in this set, and that the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(3)}(q) has radius of convergence exp(−π/\sqrt{3}) = exp(−π cot(π/3)) = 0.163033 …, and z_{(3)}(q) has |q| = 1 as natural boundary.

Case N = (4). Here, we have

F_{(4)}(z) = ∑_{n=0}^{∞} \frac{(4n)!}{(2n)! n!^{2}} z^{n} = _{2}F_{1}[\begin{matrix}1/4, 3/4 \\ 1 \end{matrix}; 64z] .

It is known (cf. [9, Theorem 2.6]) that

z_{(4)}(q) = \frac{θ_{2}^{4}(q)θ_{3}^{4}(q)}{16 (θ_{2}^{4}(q) + θ_{3}^{4}(q))^{2}} = \frac{q}{64q + ∏_{n=1}^{∞} \frac{(1 − q^{n})^{24}}{(1 − q^{2n})^{24}}} = \frac{1}{64 + \frac{H^{24}(q)}{H^{24}(q^{2})}} .  (9.6)

(The second equality can again be proven by arguments as in [15, Sections 3 and 4].)

Again alternatively, it is well-known that z_{(4)}(q) is solution of the equation (cf. [24, Sec. 5.1, p. 176])

1728J(q) = \frac{(1 + 192z_{(4)}(q))^{3}}{z_{(4)}(q)(1 − 64z_{(4)}(q))^{2}} .

By Lemma 4.1 (with M = 4, L = 4, R = 3/4) and Lemma 4.6, we know that z_{(4)}(q) has a pole of order 2 at q = − exp(−π). Hence, by (9.6), q = − exp(−π) must be a zero of 64 + \frac{H^{24}(q)}{H^{24}(q^{2})} of order 2. The function f(τ) := 64 + \frac{H^{24}(exp(2iπτ))}{H^{24}(exp(4iπτ))} is a modular form of weight 0 for Γ_{0}(2). It is easy to see that the orbit of i under SL_{2}(ℤ) splits under the action of Γ_{0}(2) into two orbits: one orbit containing i and another orbit containing δ_{1} = −\frac{1}{2} + \frac{i}{2}. The reader should note that exp(2iπδ_{1}) = − exp(−π). If we now apply the valence formula (9.2) to f(τ), then, since | stab_{δ_{1}}(Γ̂_{0}(2))| = 2, while | stab_{i}(Γ̂_{0}(2))| = 1, ord_{∞}(f) = −1, ord_{V ∞}(f) = ord_{0}(f) = 0, and | stab_{i}(Γ̂_{0}(2))| = 1, we obtain

−1 + \frac{1}{2} ord (f, δ_{1}) + ∑_{ζ∈𝔽\\{δ_{1},∞,V ∞\}} ord(f, ζ) = 0.

We know that f is analytic in the upper half plane and that ord(f, δ_{1}) = 2. Therefore, if the above equation wants to hold then necessarily all other orders must be zero.

In summary, the above arguments prove that z_{(4)}(q) is meromorphic on the unit disk |q| < 1 with poles of second order at all points exp(2iπτ), where τ is an element of the orbit of δ_{1} under Γ_{0}(2). It is not difficult to prove that exp(2iπδ_{1}) = − exp(−π) is the element of smallest modulus in this set, and that the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(4)}(q) has radius of convergence exp(−π) = exp(−π cot(π/4)) = 0.0432139 …, and z_{(4)}(q) has |q| = 1 as natural boundary.

Case N = (6). Here, we have

F_{(6)}(z) = ∑_{n=0}^{∞} \frac{(6n)!}{(3n)! (2n)! n!}z^{n} = _{2}F_{1}[\begin{matrix}1/6, 5/6 \\ 1 \end{matrix}; 432z] .

It is well-known that z_{(6)}(q) is solution of the equation (cf. [24, Sec. 5.1, p. 176])

1728J(q) = \frac{1}{z_{(6)}(q)(1 − 432z_{(6)}(q))}.  (9.7)

By Lemma 4.1 (with M = 6, L = 6, R = 5/6) and Lemma 4.6, we know that z_{(6)}(q) has an algebraic branch point with exponent −3/2 at q = − exp(−π\sqrt{3}). More precisely, since, by solving (9.7), we have

z_{(6)}(q) = \frac{1}{864} (1 − \sqrt{\frac{J(q) − 1}{J(q)}}) ,

the only singularities of z_{(6)}(q) in the interior of the unit disk can occur at points where J(q) = 0 or J(q) = 1. These are points q where Q(q) = 0, respectively where R(q) = 0. The corresponding values of q are q = exp(2iπτ), where τ runs through the elements of the orbit of ρ under SL_{2}(ℤ), respectively the elements of the orbit of i under SL_{2}(ℤ). However, since R(q) appears as a square in the definition of J(q), each point q = exp(2iπτ), where τ is in the orbit of i, is a zero of even order of J(q) − 1, whence \sqrt{J(q) − 1} is analytic at these points.

In summary, the singularities of z_{(6)}(q) in the interior of the unit disk are q = exp(2iπτ), where τ runs through the elements of the orbit of ρ under SL_{2}(ℤ). Each of them is a branch point. It is not difficult to prove that exp(2iπρ) = − exp(−π\sqrt{3}) is the element of smallest modulus in this set, and that the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(6)}(q) has radius of convergence exp(−π\sqrt{3}) = exp(−π cot(π/6)) = 0.00433342 …, and z_{(6)}(q) has |q| = 1 as natural boundary.

Case N = (2, 2). Here, we have

F_{(2,2)}(z) = ∑_{n=0}^{∞} \frac{(2n)!^{2}}{n!^{4}} z^{n} = _{2}F_{1}[\begin{matrix}1/2, 1/2 \\ 1 \end{matrix}; 16z] .

It is well-known (cf. [9, Theorem 2.2]) that

z_{(2,2)}(q) = \frac{θ_{2}^{4}(q)}{16 θ_{3}^{4}(q)} = q ∏_{n=1}^{∞} \frac{(1 − q^{4n})^{8}}{(1 − (−q)^{n})^{8}} = e^{iπ/3} \frac{H^{8}(q^{4})}{H^{8}(−q)}.  (9.8)

(In this case, the second equality results upon minor simplification from the product expressions for θ_{2}(q) and θ_{3}(q).)

Alternatively, z_{(2,2)}(q) is solution of the equation (cf. [24, Sec. 5.1, p. 176])

1728J(q) = \frac{(1 + 224z_{(2,2)}(q) + 256z_{(2,2)}^{2}(q))^{3}}{z_{(2,2)}(q)(1 − 16z_{(2,2)}(q))^{4}} .  (9.9)

From (9.8), it is evident that the radius of convergence of z_{(2,2)}(q) is 1 = exp(−π cot(π/2)), and that z_{(2,2)}(q) has |q| = 1 as natural boundary. We remark that this is consistent with Lemma 4.1.(iii) and Lemma 4.6.(iii), which imply that z_{(2,2)}(q) has an essential singularity at q = − exp(−π cot(π/2)) = −1.

9.3. Φ = 3. In this case, according to Theorem 1.2.(ii), the point z = 1/C is always a singularity of q_{N}(z). Furthermore, because of Lemma 4.2.(ii) (with d = 0), this singularity is of square root type. Consequently, according to Lemma 4.3 (with d = 0), the mirror map z_{N}(q) is analytic at q_{N}(1/C), with z′_{N}(q_{N}(1/C)) = 0.

Case N = (2, 3). Here, we have

F_{(2,3)}(z) = ∑_{n=0}^{∞} \frac{(2n)! (3n)!}{n!^{5}}z^{n} = _{3}F_{2}[\begin{matrix}1/2, 1/3, 2/3 \\ 1, 1 \end{matrix}; 108z] = _{2}F_{1}[\begin{matrix}1/3, 1/6 \\ 1 \end{matrix}; 108z]^{2} .

The identity between the hypergeometric series is a special case of Clausen’s formula (cf. [33, (2.5.7)])

3F_{2}[\begin{matrix}2a, 2b, a + b \\ 2a + 2b, a + b + \frac{1}{2} \end{matrix}; z] = _{2}F_{1}[\begin{matrix}a, b \\ a + b + \frac{1}{2} \end{matrix}; z]^{2} .  (9.10)

It is well-known (cf. e.g. [40, (23)]) that z_{(2,3)}(q) = (Q_{3}^{3}(q)−R_{3}^{2}(q))/(108Q_{3}^{3}(q)), where Q_{3}(q) and R_{3}(q) are defined as in the case N = (3). From the considerations concerning the zeroes of Q_{3}(q) and R_{3}(q) in the case N = (3), we conclude that z_{(2,3)}(q) has poles of order 6 at all points exp(2iπτ), where τ is an element of the orbit of ρ_{1} = \frac{1}{2} + \frac{i}{2\sqrt{3}} under Γ_{0}(3). The point exp(2iπρ_{1}) = − exp(−π/\sqrt{3}) is the element of smallest modulus in this set, and the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(2,3)}(q) has radius of convergence exp(−π/\sqrt{3}) = exp(−π cot(π/3)) = 0.163033 …, and z_{(2,3)}(q) has |q| = 1 as natural boundary.

Case N = (2, 4). Here, we have

F_{(2,4)}(z) = ∑_{n=0}^{∞} \frac{(4n)!}{n!^{4}}z^{n} = _{3}F_{2}[\begin{matrix}1/2, 1/4, 3/4 \\ 1, 1 \end{matrix}; 256z] = _{2}F_{1}[\begin{matrix}1/8, 3/8 \\ 1 \end{matrix}; 256z]^{2} ,

again with Clausen’s formula (9.10) explaining the identity between the hypergeometric series. It is well-known (cf. e.g. [40, (23)]) that

z_{(2,4)}(q) = (Q_{2}^{3}(q) − R_{2}^{2}(q))/(256Q_{2}^{3}(q)),  (9.11)

where Q_{2}(q) = (Q(q) + 4Q(q^{2}))/5 and R_{2}(q) = (R(q) + 8R(q^{2}))/9. An alternative expression can be found, if one observes that, by the quadratic transformation formula (see [6, Ex. 4.(iii), p. 97])

2F_{1}[\begin{matrix}a, b \\ \frac{1}{2} + a + b \end{matrix}; z] = _{2}F_{1}[\begin{matrix}2a, 2b \\ \frac{1}{2} + a + b \end{matrix}; \frac{1 − \sqrt{1 − z}}{2} ] ,  (9.12)

we have

2F_{1}[\begin{matrix}1/8, 3/8 \\ 1 \end{matrix}; 256z] = _{2}F_{1}[\begin{matrix}1/4, 3/4 \\ 1 \end{matrix}; \frac{1 − \sqrt{1 − 256z}}{2} ] ,

which enables us to identify \frac{1}{2}(1 − \sqrt{1 − 256z_{(2,4)}(q)}) with 64z_{(4)}(q), or, explicitly,

z_{(2,4)}(q) = z_{(4)}(q) − 64z_{(4)}^{2}(q).  (9.13)

By Lemma 4.1 (with M = 4, L = 4, R = 1/2) and Lemma 4.6, we know that z_{(2,4)}(q) has a pole of order 4 at q = − exp(−π). We can now either use the expression (9.11) in combination with arguments as in the case N = (4), or (9.13) and similar arguments, to conclude that z_{(2,4)}(q) is meromorphic on the unit disk |q| < 1 with poles of fourth order at all points exp(2iπτ), where τ is an element of the orbit of δ_{1} = −\frac{1}{2} + \frac{i}{2} under Γ_{0}(2). The point exp(2iπδ_{1}) = − exp(−π) is the element of smallest modulus in this set, and the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(2,4)}(q) has radius of convergence exp(−π) = exp(−π cot(π/4)) = 0.0432139 …, and z_{(2,4)}(q) has |q| = 1 as natural boundary.

We remark that, as a by-product, we obtain that Q_{2}(q) has a zero of order 2 and R_{2}(q) has a simple zero at q = − exp(−π).

Case N = (2, 6). Here, we have

F_{(2,6)}(z) = ∑_{n=0}^{∞} \frac{(6n)!}{(3n)! n!^{3}} z^{n} = _{3}F_{2}[\begin{matrix}1/2, 1/6, 5/6 \\ 1, 1 \end{matrix}; 1728z] = _{2}F_{1}[\begin{matrix}1/12, 5/12 \\ 1 \end{matrix}; 1728z]^{2} ,

again with Clausen’s formula (9.10) behind the identity between the hypergeometric series. It is well-known (cf. [25, Sec. 2]) that z_{(2,6)}(q) = 1/(1728J(q)). By the considerations in the case where N = (6), we conclude that z_{(2,6)}(q) has a pole of order 3 at all points q = exp(2iπτ), where τ runs through the elements of the orbit of ρ under SL_{2}(ℤ). The point exp(2iπρ) = − exp(−π\sqrt{3}) is the element of smallest modulus in this set, and the other elements become dense near the boundary |q| = 1. In particular, the Taylor expansion of z_{(2,6)}(q) has radius of convergence exp(−π\sqrt{3}) = exp(−π cot(π/6)) = 0.00433342 …, and z_{(2,6)}(q) has |q| = 1 as natural boundary.

Case N = (2, 2, 2). Here, we have

F_{(2,2,2)}(z) = ∑_{n=0}^{∞} \frac{(2n)!^{3}}{n!^{6}} z^{n} = _{3}F_{2}[\begin{matrix}1/2, 1/2, 1/2 \\ 1, 1 \end{matrix}; 64z] = _{2}F_{1}[\begin{matrix}1/4, 1/4 \\ 1 \end{matrix}; 64z]^{2} ,

again with Clausen’s formula (9.10) behind the identity between the hypergeometric series. By applying (9.12), we can relate the _{2}F_{1}-series to the _{2}F_{1}-series from the case where N = (2, 2). Namely, we have

2F_{1}[\begin{matrix}1/4, 1/4 \\ 1 \end{matrix}; 64z] = _{2}F_{1}[\begin{matrix}1/2, 1/2 \\ 1 \end{matrix}; \frac{1 − \sqrt{1 − 64z}}{2} ] ,

which enables us to identify \frac{1}{2}(1 − \sqrt{1 − 64z_{(2,2,2)}(q)}) with 16z_{(2,2)}(q), or, explicitly,

z_{(2,2,2)}(q) = z_{(2,2)}(q) − 16z_{(2,2)}^{2}(q).  (9.14)

The expression on the right-hand side of (9.14) can be rewritten in the form

z_{(2,2,2)}(q) = q ∏_{n=1}^{∞} \frac{(1 − q^{2n})^{24}}{(1 − (−q)^{n})^{24}} = −\frac{H^{24}(q^{2})}{H^{24}(−q)}.  (9.15)

(This follows again by arguments as in [15, Sections 3 and 4].) From (9.15), it is evident that the radius of convergence of z_{(2,2,2)}(q) is 1=exp(−π cot(π/2)), and that z_{(2,2,2)}(q) has |q| = 1 as natural boundary. We remark that this is consistent with Lemma 4.1.(iii) and Lemma 4.6.(iii), which imply that z_{(2,2,2)}(q) has an essential singularity at q = − exp(−π cot(π/2)) = −1.

10. Proof of Proposition 1.6

By Theorem 1.2.(i), the radius of convergence of q_{N}(z) is 1/C. Moreover, by Theorem 1.1, the Taylor coefficients of q_{N}(z) at 0 are all positive. Hence, q′_{N}(z) > 0 for all z in the segment [0, 1/C). By the inverse function theorem, q_{N}(z) has a compositional inverse in a domain containing [0, 1/C). This implies the first assertion.

The additional assertion on the analytic nature of z_{N}(q) around q_{N}(1/C) results from Lemma 4.2.(ii), (iii). Strictly speaking, the lemma makes no assertion about the angle (depending on ε) of the slit neighbourhood at 1/C. However, it is not very difficult to see that the assertion that there exists such a slit neighbourhood with an arbitrary ε > 0 can be derived by an appropriate refinement of the proof of Lemma 4.2.

The argument for the assertion on the analytic nature of z_{N}(q) around the point − exp ( − π cot(π/M)) given in the second paragraph after Conjecture 1.9 is similar. It is based on the singular expansion of q_{N}(z) at z = ∞ given in Lemma 4.1 and its consequence for the singular expansion of z_{N}(q) at − exp ( − π cot(π/M)) given in Lemma 4.6.

0 1/C C_{1} C_{∘}  C_{+} C_{−}

Figure 2. The contour C

11. What do we need to prove Conjecture 1.7?

By the Lagrange inversion formula (cf. [17, Theorem 1.9b with R(x) = x]), we can express the m-th Taylor coefficient of z_{N}(q) in the form

\frac{1}{2iπm} ∫_{D} \frac{dz}{q_{N}^{m}(z)},  (11.1)

where D is a sufficiently small contour that encircles the origin once in positive (that is, counter-clockwise) direction. We now deform D to a contour C consisting of four parts,

C = C_{1} ∪ C_{+} ∪ C_{−} ∪ C_{∘},

where (1) C_{1} is a (small) piece of length ℓ, say, that passes through 1/C,

(2) C_{∘} is a segment {Re^{iθ} : δ ≤ θ ≤ 2π − δ} of a (large) circle of radius R, for some δ > 0,

(3) C_{+} is a path that connects the ends of C_{1} and C_{∘} with positive real parts,

(4) C_{+} is a path that connects the ends of C_{1} and C_{∘} with negative real parts,

(5) |q_{N}(z)| ≥ q_{N}(1/C) for all z ∈ C, if there exists such a contour (see Figure 2 for a symbolic illustration of such a contour). There is no problem with the existence of C_{1} since Lemma 4.2.(ii), (iii) says that the coefficient of 1 − Cz in the singular expansion of q_{N}(z) about z = 1/C must be negative, nor is there with the existence of C_{∘}, the latter due to Theorem 1.3 and Lemma 8.1. The problem here is the question of existence of a suitable path C_{+} connecting a point in the neighbourhood of q_{N}(1/C) with a point “far out” (these points being indicated by circles in Figure 2) so that, along the path, all values of q_{N}(z) are larger in modulus than q_{N}(1/C). There cannot be any doubt that, if Φ ≥ 4, such a path exists, but, unfortunately, we have not been able to establish this. It is easy to see that (by exploiting that q_{N}(\overline{z}) = \overline{q_{N}(z)}), once we know that a suitable path C_{+} exists, there is as well a suitable path C_{−}.

Let us continue under the assumption that paths C_{+} and C_{−} as described above exist. Then we may estimate

|\frac{1}{2iπm} ∫_{D} \frac{dz}{q_{N}^{m}(z)}| = |\frac{1}{2iπm} ∫_{C} \frac{dz}{q_{N}^{m}(z)}| ≤ \frac{b}{m · q_{N}^{m}(1/C)},

where b is some constant. Since the contour integral (11.1) gives the coefficient of q^{m} in the Taylor expansion of z_{N}(q), this implies that the radius of convergence of the series z_{N}(q) must be at least q_{N}(1/C). From Theorem 1.4.(ii) it then follows that it must be exactly q_{N}(1/C).

Whether or not q_{N}(1/C) is the only point of singularity of z_{N}(q) on the boundary of its disk of convergence would have to be decided by an additional argument.

12. Conjecture 1.8 implies the first assertion in Conjecture 1.7

Let Φ ≥ 4. Conjecture 1.8 says that all but a finite number of coefficients in the Taylor series expansion of z_{N}(q) are negative. Let this Taylor series expansion be ∑_{m=0}^{∞} z_{m}q^{m}. A well-known theorem of Pringsheim (cf. [14, Theorem IV.6]) says that, if a function f(z) can be represented in a neighbourhood of z = 0 as a power series with non-negative coefficients with radius of convergence R, then f(z) has a singularity at z = R. If we apply this theorem to − ∑_{m=4}^{∞} z_{m}q^{m}, then we conclude that the radius of convergence of z_{N}(q) is a positive real number which is at the same time a singularity of z_{N}(q). By Proposition 1.6, we know that z_{N}(q) is analytic at all points q ∈ [0, q_{N}(1/C)). Thus, we know that the radius of convergence of z_{N}(q) must be at least q_{N}(1/C). On the other hand, by Theorem 1.4 we know that it cannot be larger, whence the conclusion. □

13. Conjecture 1.7 implies a weak version of Conjecture 1.8.(ii) and (iii)

Here we prove that Conjecture 1.7 implies that, if Φ_{N} ≥ 4, almost all coefficients in the Taylor expansion of z_{N}(q) at q = 0 are negative.

Proposition 13.1. Let N_{1}, N_{2}, …, N_{k} be positive integers, all at least 2, such that Φ_{N} ≥ 4, and assume that Conjecture 1.7 holds. Then there exists an A_{N} > 0 such that the coefficient of q^{m} in the Taylor series of z_{N}(q) is negative for every m > A_{N}.

Proof. By Lemma 4.2.(ii), (iii), we know that q_{N}(z) admits a singular expansion at z = 1/C of the form

q_{N}(z) = q_{N}(1/C) + q_{1}(1 − Cz) + q_{2}(1 − Cz)^{2} + ⋯

+ q_{d−1}(1 − Cz)^{d−1} + Q_{d}(1 − Cz)^{d}L̃(z) + O((1 − Cz)^{d}), where d = ⌊\frac{Φ−2}{2}⌋, q_{1} < 0, and (−1)^{d+1}Q_{d} > 0, where

L̃(z) = \begin{cases}(1 − Cz)^{\frac{1}{2}} & if Φ is odd, \\ − log(1 − Cz) & if Φ is even. \end{cases}

By Lemma 4.3, respectively Lemma 4.4, it follows that z_{N}(q) admits a singular expansion at q = q_{N}(1/C) of the form

z_{N}(q) = \frac{1}{C} + z_{1}(q − q_{N}(1/C)) + z_{2}(q − q_{N}(1/C))^{2} + ⋯

+ z_{d−1}(q − q_{N}(1/C))^{d−1} + Z_{d}(q − q_{N}(1/C))^{d}L̂(q) + O((q − q_{N}(1/C))^{d+1}), (13.1)

where d = ⌊\frac{Φ−2}{2}⌋, z_{1} = −1/(Cq_{1}), and Z_{d} > 0, where

L̂(z) = \begin{cases}(q_{N}(1/C) − q)^{\frac{1}{2}} & if Φ is odd, \\ − log (q_{N}(1/C) − q) & if Φ is even. \end{cases}

If we now use Conjecture 1.7, then the standard theorems of singularity analysis (see [14, Ch. VI]) imply that the term in (13.1) containing L̂(q) contributes the main term to the asymptotics of the Taylor coefficients of z_{N}(q). More precisely, for m → ∞, the m-th Taylor coefficient of z_{N}(q) is equal to

−const. × \frac{1 + o(1)}{q_{N}^{m}(1/C) m^{\frac{Φ}{2}}} ,  (13.2)

where const. is a positive constant which can be computed explicitly in terms of Z_{d} and Φ. The claim of the proposition is now obvious. □

Acknowledgements

The authors thank Bruce Berndt and Heng-Huat Chan for very helpful correspondence concerning the discussion of modular mirror maps presented here in Section 9.

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C. Krattenthaler, Fakultät für Mathematik, Universität Wien, Nordbergstrasze 15,

A-1090 Vienna, Austria. WWW: http://www.mat.univie.ac.at/~kratt.

T. Rivoal, Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France.

WWW: http://www-fourier.ujf-grenoble.fr/~rivoal.