ARITHMETIC PROPERTIES OF THE TAYLOR COEFFICIENTS OF DIFFERENTIALLY ALGEBRAIC POWER SERIES

C. KRATTENTHALER^{†} AND T. RIVOAL

Abstract. Let f = ∑_{n=0}^{∞} f_{n}x^{n} ∈ \overline{ℚ}[[x] be a solution of an algebraic differential equation Q(x, y(x), …, y^{(k)}(x)) = 0, where Q is a multivariate polynomial with coefficients in \overline{ℚ}. The sequence (f_{n})_{n≥0} satisfies a non-linear recurrence, whose expression involves a polynomial M of degree s. When the equation is linear, M is its indicial polynomial at the origin. We show that when M is split over ℚ, there exist two positive integers δ and ν such that the denominator of f_{n} divides δ^{n+1}(νn+ν)!^{2s} for all n ≥ 0, generalizing a well-known property when the equation is linear. This proves in this case a strong form of a conjecture of Mahler that Pólya–Popken’s upper bound n^{O(n log(n))} for the denominator of f_{n} is not optimal. This also enables us to make Sibuya and Sperber’s bound |f_{n}|_{v} ≤ e^{O(n)}, for all finite places v of \overline{ℚ}, explicit in this case. Our method is completely effective and rests upon a detailed p-adic analysis of the above mentioned non-linear recurrences. Finally, we present various examples of differentially algebraic functions for which the associated polynomial M is split over ℚ, among which are Weierstraß’ elliptic ℘ function, solutions of Painlevé equations, and Lagrange’s solution to Kepler’s equation.

In the memory of Bernard Malgrange

1. Introduction

We denote by \overline{ℚ} the field of algebraic numbers and by O\overline{ℚ} the ring of algebraic integers. Let us consider a non-trivial algebraic differential equation

Q(x, y(x), …, y^{(k)}(x)) = 0,  (1.1)

where Q ∈ ℂ[X, Y_{0}, …, Y_{k}]; here and in the sequel, by “non-trivial” we mean that Q is of degree ≥ 1 in at least one of the indeterminates Y_{j}. Let us assume that (1.1) has a power series solution f(x) := ∑_{n=0}^{∞} f_{n}x^{n} ∈ ℂ[[x]]. We say that f is a differentially algebraic power series, and write DA in short. For instance, the Taylor expansions at x = 0 of x/ log(1 + x) and of tan(x) are DA power series, as these functions are solutions of the

Date: February 14, 2025. 2020 Mathematics Subject Classification. Primary 11B37; Secondary 05A10 11B65 30B10. Key words and phrases. Non-linear recurrences, differentially algebraic power series, Gregory coeffi-

cients, Bernoulli numbers, Weierstraß ℘ function, Painlevé equations, elliptic modular invariant, Kepler’s equation, dilogarithm.

†Research partially supported by the Austrian Science Foundation FWF, grant 10.55776/F1002, in the framework of the Special Research Program “Discrete Random Structures: Enumeration and Scaling Limits”.

equations x(1 + x)y′ + y^{2} − (1 + x)y = 0 and y′ − y^{2} − 1 = 0, respectively, which are both Riccati equations.

A natural question arises: is it possible to bound |f_{n}| non-trivially from above and from below? In the Archimedean case, an upper bound was first obtained by Maillet [16], and made more precise by various authors, in particular by Malgrange [17], and in the linear case by Perron [18]. It is recalled in [15, p. 200] in the following form: there exist α, β > 0 that depend on Q and f such that |f_{n}| ≤ αn!^{β} for all n ≥ 0. A lower bound was obtained by Popken [20] under the additional assumption that f_{n} ∈ \overline{ℚ} for all n ≥ 0, when \overline{ℚ} is embedded into ℂ: there exist α, β > 0 that depend on Q and f such that either f_{n} = 0 or |f_{n}| ≥ αn^{−βn log(n)} for all n ≥ 1, and if Q is a linear polynomial in Y_{0}, …, Y_{k} (i.e., the differential equation is linear inhomogeneous with coefficients in ℂ[X]), then either f_{n} = 0 or |f_{n}| ≥ αn^{−βn} for all n ≥ 1. Similar lower bounds had been obtained earlier by Pólya [19] when f_{n} ∈ ℚ.

Popken’s lower bound has applications in the Theory of Transcendental Numbers, in particular to the values of Weierstraß’ ℘ function; see [15, pp. 207–212]. It was reproved in detail by Mahler in [15, Chapter 8], following the method used by Pólya [19] in the rational case. There it is shown to be a simple consequence of the following result. Let d_{n} denote the least positive integer such that d_{n}f_{n} is in O\overline{ℚ}. We shall from now on say that d_{n} is “the denominator of” f_{n}. Then there exist α, β > 0 that depend on Q and f such that

d_{n} ≤ αn^{βn log(n)}, n ≥ 1,  (1.2)

and if Q is a linear polynomial in Y_{0}, …, Y_{k} then d_{n} ≤ αn^{βn} for all n ≥ 1. Popken’s theorem follows by combining Maillet’s upper bound for the Galois conjugates σ(f_{n}) of f_{n} (because ∑_{n} σ(f_{n})x^{n} is also DA) and the upper bound (1.2) for d_{n}, and then by considering the norm of d_{n}f_{n} over ℚ, which is a positive integer when f_{n} ≠ 0.

Moreover, Mahler [15, p. 212] conjectured in rather loose terms that the exponent βn log(n) in Popken’s lower bound “can probably be improved to something like” βn log log(n). This would make the general bound closer to the bound in the linear case. Such an improvement, which could be difficult to detect numerically, would follow immediately from the same improvement on the exponent of n^{n} in (1.2) because in Maillet’s upper bound the exponent is even O(1). A different type of improvement of (1.2) was made by Sibuya and Sperber in [24, p. 112, Eq. (III’)], who proved that, for all finite places v of \overline{ℚ}, |f_{n}|_{v} ≤ e^{cn} for some non-explicit constant c > 0 that depends on f, Q, and v; this had been conjectured by Dwork. This proves in particular that f has positive v-adic radius of convergence in ℂ_{p}. Earlier, Mahler [14] had proved a lower bound for |f|_{v} similar to Popken’s in the Archimedean case. We refer to [23] for a survey of Archimedean and non-Archimedean estimates for f_{n} when f is a DA power series, in particular for (possibly not optimal) explicit values of the constants α and β.

Mahler’s denominator conjecture was the starting point of our investigations, and this also led us naturally to study Sibuya and Sperber’s v-adic bounds. Let us first consider two simple examples in the linear case, both of hypergeometric type. The series h_{1}(x) := ∑_{n=0}^{∞} \frac{x^{k}}{∏_{k=0}^{n−1}(k^{2}+1)} is a DA power series solution to x^{2}y′′′ + xy′′ + (1 − x)y′ − y = 0, and the associated denominator d_{n} satisfies d_{n} = ∏_{k=0}^{n−1}(k^{2} + 1) ≤ ∏_{k=0}^{n−1}(k + 1)^{2} = n!^{2}, as expected. Notice however that, conjecturally, there exist no integers δ, ν, μ, s ≥ 0 such that d_{n} divides δ^{n+1}(νn + μ)!^{s} for all n ≥ 0 (^{1}). On the other hand, the series h_{2}(x) := ∑_{n=0}^{∞} \frac{x^{k}}{∏_{k=0}^{n−1}(2k+1)} is also a DA power series solution to 2xy′′+(1−x)y′−y = 0. Here, the associated denominator d_{n} = ∏_{k=0}^{n−1}(2k + 1) is not only bounded by ∏_{k=0}^{2n−1}(k + 1) = (2n)!, but this time it divides (2n)!. This is not surprising: it is well-known that such a divisibility phenomenon holds for all power series in \overline{ℚ}[[x]] that are solution of a homogeneous linear differential equation with coefficients in \overline{ℚ}[x] and whose indicial polynomial at the origin has only rational roots. This is the case for h_{2} (roots 0 and 1/2) but not for h_{1} (roots 0 and 1 ± i). See §3.1 for the details.

A little more surprising may be the fact that a divisibility phenomenon also numerically holds for certain DA series that are not necessarily solutions of a linear differential equation. Let us consider for instance the series f(x) = 1 + ∑_{n=1}^{∞} f_{n}x^{n} that satisfies the Riccati equation xf′(x) − xf(x)^{2} + af(x) − a = 0, where a ≥ 1 is a fixed integer. The sequence (f_{n})_{n≥0} satisfies the non-linear recurrence relation

f_{n+1} = \frac{1}{n + a + 1} ∑_{j=0}^{n} f_{j}f_{n−j}, n ≥ 0, f_{0} := 1,  (1.3)

and we observed numerically that seemingly n! (n + a)! f_{n} ∈ ℤ for all n ≥ 0. (Note that, if a = 0, then f_{n} = 1 for all n ≥ 0.) Our first thought was that this observation would be easy to prove by considering the sequence φ_{n} := n! (n + a)! f_{n}, which is solution of the recurrence

φ_{n+1} = ∑_{j=0}^{n} \frac{n + 1}{(j + 1)_{a}} \binom{n}{j}\binom{n + a}{j}φ_{j}φ_{n−j}, n ≥ 0, φ_{0} := a!,

where the Pochhammer symbol (α)_{m} is defined by α(α + 1) ⋯ (α + m − 1) for m ≥ 1, and (α)_{0} = 1. The binomial coefficients are obviously an important gain but this transformation comes with a new “big” denominator (j +1)_{a}. By a careful p-adic analysis of the summand (see Proposition 1 in §2), we shall prove that δ^{n+1}φ_{n} ∈ ℤ for all n ≥ 0 for some integer δ ≥ 1. The proof of this arithmetic property, which is weaker than what seems to be true, is much more complicated than what one might expect at first sight. On the other hand, this yields a better result for f_{n} than the upper bound (1.2), i.e., we have δ^{n+1}n! (n+a)! f_{n} ∈ ℤ for all n ≥ 0, and it even proves Mahler’s conjecture in this case in a stronger form. This also makes the upper bound for |f|_{v} by Sibuya and Sperber explicit.

1Assume on the contrary the existence of such integers. Then, any prime divisor p of d_{n} divides either δ or (νn+μ)!, hence p ≤ max(δ, νn+μ). But it is widely believed, because it is an instance of Bouniakovsky’s conjecture, that there exist infinitely many primes of the form m^{2} + 1 where m ∈ ℕ, so that we would have (n − 1)^{2} + 1 ≤ max(δ, νn + μ) for infinitely many n. This is not possible. See [22] and references therein for estimates of the p-adic valuation of products of the form ∏_{1≤k≤n} |q(k)|, where q is a polynomial of degree ≥ 2, irreducible over ℚ, and taking integer values at integer arguments.

To explain our contribution towards Mahler’s denominator conjecture, we start by reviewing properties proved by Mahler in [15, Chapter 8]. We assume that the differential equation (1.1) has a solution f(x) := ∑_{n=0}^{∞} f_{n}x^{n} ∈ \overline{ℚ}[[x]]. Then, by [15, p. 186, §120], the differential equation can be assumed to have coefficients in \overline{ℚ} without loss of generality. Mahler proved in [15, p. 194] that the sequence of Taylor coefficients (f_{n})_{n≥0} satisfies a non-linear recurrence

f_{n+1} = \frac{1}{M(n)} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} _{∑_{j_{1}+···+j_{k}=n−σ} ; 0≤j_{1},...,j_{k}≤n}P_{σ,k}(n, j_{1}, j_{2}, …, j_{k})f_{j_{1}}f_{j_{2}} ⋯ f_{j_{k}}, for n ≥ N, (1.4)

where N is some non-negative integer, M(X) ∈ O\overline{ℚ}[X] vanishes for no n ≥ N, the coefficients P_{σ,k}(n, j_{1}, j_{2}, …, j_{k}) are in O\overline{ℚ}, the constants σ_{1}, σ_{2} are integers, s and k_{0} are positive integers. More precisely, the P_{σ,k}(n, j_{1}, j_{2} …, j_{k}) are piecewise polynomials in n, j_{1}, j_{2}, …, j_{k} with coefficients in O\overline{ℚ}. The initial values f_{0}, f_{1}, …, f_{N} are algebraic numbers with common denominator D. It should be observed that the restriction j_{1}, …, j_{k} ≤ n can be safely ignored for non-negative σ, while it has an effect for negative σ. The process to determine explicitly the recurrence (1.4) from the polynomial Q can be complicated in general but it is effective; it is a task that can be performed easily for simple examples. Following Pólya in the rational case, Mahler then proved the upper bound on the denominator of f_{n} as follows: he showed by induction on n that Δ_{n} := ∏_{j=N}^{n} |A(j)|^{⌊ \frac{(m−1)n+1}{(m−1)j+1} ⌋} is a denominator of f_{n} for n ≥ N, where A(X) ∈ ℤ[x] vanishes for no integer n ≥ N and is constructed from M in [15, p. 202], and where the integer m ≥ 1, which depends only on Q and is defined in [15, p. 187], is equal to 1 if and only if Q is linear in Y_{0}, …, Y_{k}. The denominator d_{n} of f_{n} divides Δ_{n}, which then has to be bounded. When m = 1, the exponent of each A(j) in the product for Δ_{n} is always 1 and thus independent of n, while if m ≥ 2 it depends on n. This explains the difference between the upper bounds for d_{n} in the linear and non-linear cases.

In this paper, we shall consider the particular situation where the polynomial M in (1.4) is split over ℚ, i.e., the sequence of Taylor coefficients (f_{n})_{n≥0} satisfies a non-linear recurrence of the form

f_{n+1} =

\frac{1}{C ∏_{i=1}^{s}(a_{i}n + b_{i})} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} _{∑_{j_{1}+···+j_{k}=n−σ} ; 0≤j_{1},...,j_{k}≤n}P_{σ,k}(n, j_{1}, j_{2}, …, j_{k})f_{j_{1}}f_{j_{2}} ⋯ f_{j_{k}}, for n ≥ N, (1.5)

where N is some non-negative integer, the coefficients P_{σ,k}(n, j_{1}, j_{2}, …, j_{k}) are in O\overline{ℚ} (^{2}), C is a non-zero integer, σ_{1}, σ_{2} are integers, s and k_{0} are positive integers, the b_{i}’s are integers, the a_{i}’s are positive integers such that gcd(a_{i}, b_{i}) = 1 and a_{i}n + b_{i} ≠ 0 for all i ∈ {1, …, s} and all n ≥ N. The initial values f_{0}, f_{1}, …, f_{N} are algebraic numbers with common denominator D. In this situation, our main result is a proof of Mahler’s denominator conjecture in a much stronger form, i.e., a divisibility property of d_{n} which in particular implies that log(n) can be replaced by O(1), not just log log(n).

2Due to the normalization adopted in (1.5), the coefficients P_{σ,k} in (1.5) might differ from the P_{σ,k} in (1.4) by a common non-zero algebraic factor (independent of σ and k). The important point for us is that we can still assume without loss of generality that they are in O\overline{ℚ}; indeed, this can always be achieved by changing C to mC for a suitable integer m ≥ 2 if necessary.

Theorem 1. Let f ∈ \overline{ℚ}[[x] be a solution of a non-trivial algebraic differential equation Q(x, y(x), …, y^{(k)}(x)) = 0, where Q ∈ \overline{ℚ}[X, Y_{0}, …, Y_{k}]. More specifically, we assume that the sequence (f_{n})_{n≥0} of Taylor coefficients of f satisfies a non-linear recurrence of the form (1.5).

Then there exist two positive integers δ and ν such that the denominator of f_{n} divides δ^{n+1}(νn + ν)!^{2s} for all n ≥ 0, where s is the degree of the denominator polynomial on the right-hand side of (1.5).

As a consequence, we obtain an explicit version of the v-adic upper bound of Sibuya and Sperber in this particular situation, which seems to be new.

Corollary 1. In the setting of Theorem 1, for all finite places v of \overline{ℚ} over any given rational prime number p, we have

|f_{n}|_{v} ≤ p^{(v_{p}(δ)+ \frac{2sν}{p−1})(n+1)}, n ≥ 0,  (1.6)

where we take the standard normalization |p|_{v} := 1/p.

(On the right-hand side of (1.6), v_{p} denotes the usual p-adic valuation, and the exponent involves ν and not the place v.) The proof of Theorem 1 provides completely explicit, albeit complicated, formulas for δ and ν in terms of C, D, k_{0}, a_{1}, a_{2}, …, a_{s} and b_{1}, b_{2}, …, b_{s}. When σ_{1} ≥ 0, these formulas, in a refined form, are given by Theorem 2 in §2; they are not always sharp. When σ_{1} < 0 for a sequence (f_{n})_{n≥0} defined for all n ≥ N by a recurrence like (1.5), we shall prove in a constructive way (Lemma 1) that we can always find another recurrence for (f_{n})_{n≥0} like (1.5) in which now σ_{1} ≥ 0, but for n ≥ Ñ which may differ from N. We shall explain in §2.1 that Lemma 1 is a variant of [23, p. 382, Lemma 2.2]. The latter also provides in principle another way of finding a recurrence of type (1.5) from a recurrence of type (1.4), even though it seems not to be entirely constructive, as we explain at the end of §2.1.

The arithmetic assumption on the polynomial M implies in particular the rationality of all the roots (known as the local exponents) of the indicial polynomial at 0 when Q is linear in Y_{0}, …, Y_{k}. The procedure in [23, p. 382] has the theoretical interest of explaining why M can be viewed as a natural generalization to the non-linear case of the indicial polynomial at the origin in the linear case. A similar assumption appears in [23, p. 400, (ii)], where Popken’s lower bound is then improved to |f_{n}| ≥ αn^{−βn} for some non-explicit positive constants α and β. It does not seem that the stronger and more precise divisibility properties in Theorem 1 and Theorem 2 below are also proven. It turns out that the arithmetic assumption holds in many interesting situations. We shall elaborate on these examples in §3.

In the next section, we present our proof of Theorem 1. It consists in several steps. In Lemma 1 we show that, if in the sum on the right-hand side of (1.5) negative σ’s contribute, then one can find an equivalent recurrence of the same type without negative σ’s contributing; we also compare it with Lemma 2.2 in [23]. We then restate Theorem 1 in a more precise, but also more general form, see Theorem 2. The key arithmetic part of its proof is taken care of separately in Proposition 1. In the last section, Section 3, we discuss many examples in which our theorems apply, amongst which are Gregory coefficients, tangent numbers, Bernoulli numbers, Taylor coefficients of the Weierstraß elliptic function, of solutions to Painlevé equations, of modular forms, of solutions to Kepler’s equation, of the compositional inverse of the dilogarithm. We conclude the paper with a discussion of Mahler’s conjecture on a simple example of a recurrence for which M is non-split over ℚ.

2. Proof of Theorem 1

2.1. An effective procedure to remove negative σ’s. In this subsection, we show that, if there are negative σ’s contributing to the right-hand side of (1.5), then there is an equivalent recurrence with all the characteristics of (1.5) but which avoids negative σ’s, i.e., it satisfies σ_{1} ≥ 0. By equivalent recurrences of type (1.5), we mean that the given sequence (f_{n})_{n≥0} satisfies both recurrences for all n large enough, but “large enough” is different in each recurrence. At the end of this subsection, we shall also mention a non-constructive variant of Lemma 1, due to Sibuya and Sperber, which nonetheless has important theoretical consequences for us.

Lemma 1. For any recurrence of the form (1.5) in which negative σ’s appear, there is an equivalent one with only non-negative σ’s.

The proof of Lemma 1 is constructive, and free of any indeterminacy. As an inspection of the proof reveals, the conclusion of Lemma 1 holds more generally also for a recurrence of type (1.4).

Proof. We consider a recurrence (1.5) in which σ_{1} is negative. Let N̄ := max{N, σ_{2}−k_{1}σ_{1}}. (The background for this choice will become clear later on.) We assume the f_{j}’s with j < N̄ as initial conditions, meaning that f_{j} with N < j < N̄ has been computed via the recurrence (1.5).

Now, in the sums over j_{t}’s, we isolate terms containing f_{j}’s with j < −σ_{1} (recall that −σ_{1} > 0) in separate sums. This converts (1.5) into

f_{n+1} = \frac{1}{C ∏_{i=1}^{s}(a_{i}n + b_{i})} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} ∑_{ℓ=0}^{k} \binom{k}{ℓ} ∑_{0≤j_{1},...,j_{ℓ}<−σ_{1}} f_{j_{1}} ⋯ f_{j_{ℓ}}

·_{∑_{j_{ℓ+1}+···+j_{k}=n−σ−j_{1}−···−j_{ℓ}} ; −σ_{1}≤j_{ℓ+1},...,j_{k}≤n}P_{σ,k}(n, j_{1}, j_{2}, …, j_{k})f_{j_{ℓ+1}} ⋯ f_{j_{k}}

= \frac{1}{C ∏_{i=1}^{s}(a_{i}n + b_{i})} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} ∑_{ℓ=0}^{k} \binom{k}{ℓ} ∑_{0≤j_{1},...,j_{ℓ}<−σ_{1}} f_{j_{1}} ⋯ f_{j_{ℓ}}

·_{∑_{j_{ℓ+1}+···+j_{k}=n−σ−j_{1}−···−j_{ℓ}+(k−ℓ)σ_{1}} ; 0≤j_{ℓ+1},...,j_{k}≤n+σ_{1}}P_{σ,k}(n, j_{1}, …, j_{ℓ}, j_{ℓ+1} − σ_{1}, …, j_{k} − σ_{1})f_{j_{ℓ+1}−σ_{1}} ⋯ f_{j_{k}−σ_{1}}.

The binomial coefficient arises since the ℓ terms f_{j} with j < −σ_{1} might appear at any of the k possible positions.

Now we write g_{m} = f_{m−σ_{1}}. In this notation, the above recurrence reads

g_{n+1+σ_{1}} = \frac{1}{C ∏_{i=1}^{s}(a_{i}n + b_{i})} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} ∑_{ℓ=0}^{k} \binom{k}{ℓ} ∑_{0≤j_{1},...,j_{ℓ}<−σ_{1}} f_{j_{1}} ⋯ f_{j_{ℓ}}

·_{∑_{j_{ℓ+1}+···+j_{k}=n−σ−j_{1}−···−j_{ℓ}+(k−ℓ)σ_{1}} ; 0≤j_{ℓ+1},...,j_{k}≤n+σ_{1}}P_{σ,k}(n, j_{1}, …, j_{ℓ}, j_{ℓ+1} − σ_{1}, …, j_{k} − σ_{1})g_{j_{ℓ+1}} ⋯ g_{j_{k}},

or, after performing the shift n ↦ n − σ_{1},

g_{n+1} = \frac{1}{C ∏_{i=1}^{s}(a_{i}n − a_{i}σ_{1} + b_{i})} ∑_{σ=σ_{1}}^{σ_{2}} ∑_{k=1}^{k_{0}} ∑_{ℓ=0}^{k} \binom{k}{ℓ} ∑_{0≤j_{1},...,j_{ℓ}<−σ_{1}} f_{j_{1}} ⋯ f_{j_{ℓ}}

·_{∑_{j_{ℓ+1}+···+j_{k}=n−σ−j_{1}−···−j_{ℓ}+(k−ℓ−1)σ_{1}} ; 0≤j_{ℓ+1},...,j_{k}≤n}P_{σ,k}(n − σ_{1}, j_{1}, …, j_{ℓ}, j_{ℓ+1} − σ_{1}, …, j_{k} − σ_{1})g_{j_{ℓ+1}} ⋯ g_{j_{k}},

for n ≥ N̄. (2.1)

Define σ̄ := σ + j_{1} + ⋯ + j_{ℓ} − (k − ℓ − 1)σ_{1}. If k > ℓ + 1, we have σ̄ ≥ σ − σ_{1} ≥ 0. This implies that in this case the inner sum in (2.1) runs over non-negative integers j_{ℓ+1}, …, j_{k} with j_{ℓ+1} +⋯+j_{k} = n−σ̄, where σ̄ ≥ 0. The case k = ℓ+1 does not need to be considered since then the inner sum consists of just one (or no term). The equality k = ℓ produces an empty sum since the condition on the summation indices in the inner sum in (2.1) (the empty sum being equal to zero) becomes

0 = n − σ − j_{1} − ⋯ − j_{ℓ} + (k − ℓ − 1)σ_{1} = n − σ − j_{1} − ⋯ − j_{k} − σ_{1}  (2.2)

in this case, but

n − σ − j_{1} − ⋯ − j_{k} − σ_{1} > N̄ − σ + kσ_{1} − σ_{1} ≥ σ_{2} − σ − σ_{1} > 0

by the definition of N̄, a contradiction with (2.2).

In summary, the recurrence (2.1) for (g_{n})_{n≥0} = (f_{n−σ_{1}})_{n≥0} is indeed a recurrence of the form (1.5) without negative σ’s; in other words, it is of the form (1.5) with σ_{1} = 0. □

As mentioned before, Sibuya and Sperber proved an important lemma that leads to the same conclusion as our Lemma 1. Starting from a solution f = ∑_{n=0}^{∞} f_{n}x^{n} ∈ \overline{ℚ}[[x]] of a non-trivial differential algebraic equation Q(x, y, …, y^{(k)}) = 0 with coefficients

8

in \overline{ℚ}, they first consider a non-trivial differential equation Q̃(x, y, …, y^{(ℓ)}) = 0 with Q̃ ∈ \overline{ℚ}[X, Y_{0}, …, Y_{ℓ}] of which f is still a solution. They require that ℓ ≥ 0 is minimal amongst all differential algebraic equations satisfied by f, and that the degree of Q̃ in Y_{ℓ} is also minimal. Notice that ℓ is the transcendence degree of the field generated over \overline{ℚ}(x) by f and all its derivatives. This in particular ensures that

\frac{∂Q̃}{∂Y_{ℓ}} (x, f, …, f^{(ℓ)}) ≠ 0,

which is crucial. However, Sibuya and Sperber give no way to effectively compute ℓ and to minimize the degree in Y_{ℓ} starting from f and Q above. They canonically attach to Q̃ and f a linear operator L_{0} ∈ \overline{ℚ}[[x]][\frac{d}{dx}] of order ℓ and then consider the indicial polynomial P_{0} at the origin of L_{0}. Then [23, Lemma 2.2] says the following:

For any c ≥ 0, there exist integers N′ ≥ 0, N ≥ N′, N′′ ≥ c such that u := ∑_{n=N}^{∞} f_{n}x^{n−N′} ∈ \overline{ℚ}[[x]] is a solution of a differential equation L(u) = x^{N′′}F (x, u, u′, …, u^{(ℓ)}) where F ∈ O\overline{ℚ}[X, Y_{0}, …, Y_{ℓ}] and L is a linear differential operator in O\overline{ℚ}[x][\frac{d}{dx}] of order ℓ and whose leading coefficient is independent of N′′.

Moreover, the indicial polynomial P (X) at the origin of L is simply P_{0}(X + N′); in particular, if P_{0} is split over ℚ, then P is as well. From the particular form of the equation satisfied by u, Sibuya and Sperber deduce a recurrence for (f_{n})_{n≥0} which is of the form (1.4) with σ_{1} ≥ 0 (provided N′′ is large enough, which is possible because c is arbitrary) and with M(X) = P_{0}(X + N′ + 1). Therefore, this gives an interpretation of M in terms of the indicial polynomial P_{0} attached to Q̃ and f. It is important to observe that P_{0} may be different for another solution in \overline{ℚ}[[x]] of the differential equation Q(x, y, …, y^{(k)}) = 0.

Sibuya and Sperber also drew a few consequences from their lemma in [23, p. 384] (encompassed by our Theorem 1) when P_{0} is a constant, when 0 is an ordinary point of L_{0}, and when F (x, u, u′, …, u^{(k)}) = F (x) (which happens when Q̃ is linear in Y_{0}, …, Y_{k}). In the latter case, they recovered Popken’s upper bound for the denominator of f_{n} in the linear case; this is in fact equivalent to what Mahler had done in [15, p. 205].

2.2. The main result. We are now in the position to state our main result, Theorem 2 below, which together with Lemma 1 immediately implies Theorem 1. Indeed, the denominator d_{n}(C, D, a, b, k_{0}) in (2.5) below obviously divides δ^{n+1}(νn + ν)!^{2s} for all n ≥ N for suitable positive integers δ and ν. We can also take δ large enough to ensure that D divides δ. Hence, the denominator d_{n} of f_{n} divides δ^{n+1}(νn + ν)!^{2s} for all n ≥ 0.

To prove Corollary 1, let v be a place of \overline{ℚ} over a rational prime number p. From the divisibility in Theorem 1, we deduce that

|f_{n}|_{v} ≤ p^{v_{p}(δ^{n+1})+v_{p}((νn+ν)!^{2s})} ≤ p^{v_{p}(δ)(n+1)+ \frac{2sν}{p−1} (n+1)},

because by Legendre’s formula [12, p. 10] for the p-adic valuation of factorials, i.e.,

v_{p}(n!) = ∑_{ℓ≥1} ⌊\frac{n}{p^{ℓ}} ⌋ ,  (2.3)

we have

v_{p}((νn + ν)!^{2s}) ≤ 2s ∑_{ℓ=1}^{∞} \frac{νn + ν}{p^{ℓ}} = \frac{2sν(n + 1)}{p − 1}.

This proves (1.6).

The proof of Theorem 2 requires a technical auxiliary result which is established separately in Proposition 1 below. We emphasize that Theorem 2 applies to more general situations than the differential context behind Theorem 1. Indeed, the only assumption we make below on the coefficients P_{σ,k}(n, j_{1}, j_{2}, …, j_{k}) is that they are algebraic integers; they need not necessarily be piecewise polynomials in n, j_{1}, j_{2}, …, j_{k} as they are when they come from a solution of an algebraic differential equation.

Theorem 2. Let us consider a recurrence of the form (1.5) without negative σ’s, that is,

f_{n+1}

= \frac{1}{C ∏_{i=1}^{s}(a_{i}n + b_{i})} ∑_{σ=0}^{σ_{2}} ∑_{k=1}^{k_{0}} _{∑_{j_{1}+···+j_{k}=n−σ} ; 0≤j_{1},...,j_{k}≤n}P_{σ,k}(n, j_{1}, j_{2}, …, j_{k})f_{j_{1}}f_{j_{2}} ⋯ f_{j_{k}}, for n ≥ N, (2.4)

where P_{σ,k}(n, j_{1}, j_{2}, …, j_{k}) are algebraic integers, σ_{2} is a non-negative integer, C, s and k_{0} are positive integers, b = (b_{1}, b_{2}, …, b_{s}) is a vector of integers, a = (a_{1}, a_{2}, …, a_{s}) is a vector of positive integers such that gcd(a_{i}, b_{i}) = 1 for i = 1, 2, …, s, and N is some non-negative integer such that a_{i}N + b_{i} − a_{i} ≥ 0 for i = 1, 2, …, s. The initial values f_{0}, f_{1}, …, f_{N} are algebraic numbers with common denominator D ≥ 1.

Then, for all n ≥ N, the denominator of f_{n} divides

d_{n}(C, D, a, b, k_{0}) := C^{n}D^{(k_{0}−1)n+1}Π^{n} ∏_{i=1}^{s}(a_{i}n)! (a_{i}n + b_{i} − a_{i})!,  (2.5)

where

Π = ∏_{i=1}^{s} (( max{b_{i} − a_{i}, 0})!^{k_{0}−1}_{∏_{p<2 max_{1≤j≤s}(b_{j}−a_{j})} p prime}p^{⌈log_{p} max{b_{i}−a_{i},1}⌉}).  (2.6)

Proof. The assertion is certainly true for n = N. For, by assumption, the denominator of f_{N} is D, which trivially divides d_{N}(C, D, a, b, k_{0}) due to the term D^{(k_{0}−1)N+1} in its definition.

For n > N we are going to argue by induction. Put

φ_{n} = d_{n}(C, D, a, b, k_{0})f_{n} = C^{n}D^{(k_{0}−1)n+1}Π^{n} (∏_{i=1}^{s}(a_{i}n)! (a_{i}n + b_{i} − a_{i})!) f_{n}, for n > N.

(2.7)

and

φ_{n} = C^{n}D^{(k_{0}−1)n+1}Π^{n}f_{n}, for 0 ≤ n ≤ N.

By multiplying both sides of the recurrence (2.4) by

d_{n+1}(C, D, a, b, k_{0}) = C^{n+1}D^{(k_{0}−1)(n+1)+1}Π^{n+1} ∏_{i=1}^{s}(a_{i}n + a_{i})! (a_{i}n + b_{i})!,

we obtain

φ_{n+1} = ∑_{σ=0}^{σ_{2}} ∑_{k=1}^{k_{0}} C^{σ}D^{(k_{0}−1)σ+k_{0}−k}Π^{σ+1}_{∑_{j_{1}+···+j_{k}=n−σ} ; 0≤j_{1},...,j_{k}≤n}( ∏_{i=1}^{s} \frac{(a_{i}n + a_{i})! (a_{i}n + b_{i} − 1)!}{_{∏_{t=1}^{k} ; j_{t}>N}(a_{i}j_{t})! (a_{i}j_{t} + b_{i} − a_{i})!})

· P_{σ,k}(n, j_{1}, j_{2}, …, j_{k})φ_{j_{1}}φ_{j_{2}} ⋯ φ_{j_{k}}, for n ≥ N, (2.8)

where (as indicated) the product over t is taken only over those t between 1 and k for which j_{t} > N.

We claim that the recurrence (2.8) is one with algebraic integer coefficients throughout. Obviously, by induction, this claim would imply that all φ_{n}’s — including those with n > N — are algebraic integers. Going back to the definitions (2.7) of φ_{n} and (2.5) of d_{n}(C, D, a, b, k_{0}), it follows that the denominator of f_{n} divides d_{n}(C, D, a, b, k_{0}) for all n ≥ 0.

Clearly, the only problem concerning our claim could arise from the product over i, a product of factorial ratios. It is Proposition 1 below which addresses this product. In order to apply the proposition, we need to reorder the j_{t}’s in increasing order (so that the τ_{i} in the proposition are well-defined). Then Proposition 1 says that the product of factorial ratios is “almost” an integer, except for some prime divisors p with p < 2 max_{1≤i≤s}(b_{i} −a_{i}). However, their product is a divisor of Π (compare its definition in (2.6) with the right-hand side of (2.10)), which in turn divides Π^{σ+1} which appears as a factor in (2.8). This proves our claim and completes the proof of the theorem. □

We now proceed with the proof of the last missing step in the proof of Theorem 2. This is the most complicated part.

Proposition 1. Let n, k, s and a_{1}, a_{2}, …, a_{s} be positive integers with k ≥ 2, let b_{1}, b_{2}, …, b_{s} be integers, and let σ, τ_{1}, τ_{2}, …, τ_{k}, and j_{1}, j_{2}, …, j_{k} be non-negative integers with j_{1} + j_{2} + ⋯ + j_{k} = n − σ and such that a_{i}j_{t} + b_{i} − a_{i} < 0 for 1 ≤ t ≤ τ_{i} and 1 ≤ i ≤ s, while a_{i}j_{t} + b_{i} − a_{i} ≥ 0 for τ_{i} + 1 ≤ t ≤ k and 1 ≤ i ≤ s, In particular, if b_{i} − a_{i} ≥ 0, then necessarily τ_{i} = 0.

(1) For prime numbers p ≥ max_{1≤i≤s} 2(b_{i} − a_{i}), we have

v_{p} (∏_{i=1}^{s} \frac{(a_{i}n + a_{i})! (a_{i}n + b_{i} − 1)!}{(a_{i}j_{τ_{i}+1})! (a_{i}j_{τ_{i}+1} + b_{i} − a_{i})! ⋯ (a_{i}j_{k})! (a_{i}j_{k} + b_{i} − a_{i})!}) ≥ 2 ∑_{i=1}^{s} v_{p}((a_{i} −1)! (a_{i}σ)!).

(2.9)

(2) For prime numbers p < max_{1≤i≤s} 2(b_{i} − a_{i}), we have

v_{p} (∏_{i=1}^{s} \frac{(a_{i}n + a_{i})! (a_{i}n + b_{i} − 1)!}{(a_{i}j_{τ_{i}+1})! (a_{i}j_{τ_{i}+1} + b_{i} − a_{i})! ⋯ (a_{i}j_{k})! (a_{i}j_{k} + b_{i} − a_{i})!})

≥ ∑_{i=1}^{s} ( − ⌈log_{p} max{b_{i} − a_{i}, 1}⌉ − (k − 1)v_{p}(max{b_{i} − a_{i}, 0}!) + 2v_{p}((a_{i} − 1)! (a_{i}σ)!)). (2.10)

Proof. For this proof, we drew inspiration from our previous work [9], in which we also proved another denominator conjecture in the context of zeta values, and not related to Mahler’s denominator conjecture studied in the present paper.

By Legendre’s formula recalled above in (2.3), we have

v_{p} (∏_{i=1}^{s} \frac{(a_{i}n + a_{i})! (a_{i}n + b_{i} − 1)!}{(a_{i}j_{τ_{i}+1})! (a_{i}j_{τ_{i}+1} + b_{i} − a_{i})! ⋯ (a_{i}j_{k})! (a_{i}j_{k} + b_{i} − a_{i})!})

= ∑_{ℓ≥1} ∑_{i=1}^{s} (⌊\frac{a_{i}n + b_{i} − 1}{p^{ℓ}}⌋ + ⌊\frac{a_{i}n + a_{i}}{p^{ℓ}} ⌋

− ⌊\frac{a_{i}j_{τ_{i}+1}}{p^{ℓ}} ⌋ − ⋯ − ⌊\frac{a_{i}j_{k}}{p^{ℓ}} ⌋ − ⌊\frac{a_{i}j_{τ_{i}+1} + b_{i} − a_{i}}{p^{ℓ}} ⌋ − ⋯ − ⌊\frac{a_{i}j_{k} + b_{i} − a_{i}}{p^{ℓ}} ⌋) . (2.11)

We put N_{i} = \{\frac{a_{i}n}{p^{ℓ}}\}, J_{t,i} = \{\frac{a_{i}j_{t}}{p^{ℓ}} \}, for t = 1, 2, …, k − 1 (sic!), A_{i} = \{\frac{a_{i}−1}{p^{ℓ}}\} and

B_{i} = \{\frac{b_{i}−a_{i}}{p^{ℓ}} \} for i = 1, 2, …, s, and S_{i} = \{\frac{a_{i}σ}{p^{ℓ}}\}. The reader should keep in mind that N_{i}, J_{t,i}, A_{i}, B_{i}, S_{i} also depend on ℓ. We do not indicate this in the notation for better readability.

Using the above notation, the summand of the sum over ℓ on the right-hand side of (2.11) can be rewritten as

∑_{i=1}^{s} ( ⌊N_{i} + A_{i} + B_{i}⌋ + ⌊N_{i} + A_{i} + \frac{1}{p^{ℓ}} ⌋ − ∑_{t=τ_{i}+1}^{k−1} ( ⌊J_{t,i}⌋ + ⌊J_{t,i} + B_{i}⌋ )

− ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ − ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} + B_{i} − S_{i}⌋

+ 2 ∑_{t=1}^{τ_{i}} ⌊\frac{a_{i}j_{t}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1 − τ_{i}) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ )

= ∑_{i=1}^{s} ( ⌊N_{i} + A_{i} + B_{i}⌋ + ⌊N_{i} + A_{i} + \frac{1}{p^{ℓ}} ⌋ − ∑_{t=τ_{i}+1}^{k−1} ⌊J_{t,i} + B_{i}⌋

− ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ − ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} + B_{i} − S_{i}⌋

+ 2 ∑_{t=1}^{τ_{i}} ⌊\frac{a_{i}j_{t}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1 − τ_{i}) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ ). (2.12)

(1) Concerning the summand, we claim that

⌊N_{i} + A_{i} + B_{i}⌋ + ⌊N_{i} + A_{i} + \frac{1}{p^{ℓ}} ⌋ − ∑_{t=τ_{i}+1}^{k−1} ⌊J_{t,i} + B_{i}⌋

− ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ − ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} + B_{i} − S_{i}⌋

+ 2 ∑_{t=1}^{τ_{i}} ⌊\frac{a_{i}j_{t}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1 − τ_{i}) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋

≥ 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ . (2.13)

For a fixed i, we consider first the case where b_{i} < a_{i}. Then we have

∑_{t=τ_{i}+1}^{k−1} ⌊J_{t,i} + B_{i}⌋ ≤ k − 1 − τ_{i} ≤ −(k − 1 − τ_{i}) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋  (2.14)

and

⌊N_{i} + A_{i} + B_{i}⌋ ≥ ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} + B_{i} − S_{i}⌋ ,  (2.15)

which establishes (2.13) in this case.

Now let b_{i} ≥ a_{i}. We note that then necessarily τ_{i} = 0. Without loss of generality we assume that, for fixed i with 1 ≤ i ≤ s, we have J_{t,i} + B_{i} ≥ 1 for 1 ≤ t ≤ m_{i} and J_{t,i} + B_{i} < 1 for m_{i} + 1 ≤ t ≤ k − 1, for some non-negative integer m_{i}.

Since by assumption p ≥ 2(b_{i} − a_{i}), we have ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ = 0 and

B_{i} = \{\frac{b_{i} − a_{i}}{p^{ℓ}} \} = \frac{b_{i} − a_{i}}{p^{ℓ}} ≤ \frac{b_{i} − a_{i}}{2(b_{i} − a_{i})} = \frac{1}{2}.

It should be observed that the same conclusions also hold for b_{i} − a_{i} = 0.

The case of m_{i} = 0 of our claim follows from (2.15).

Let now m_{i} ≥ 1. If N_{i} + B_{i} < 1, then

N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < 1 − \frac{m_{i}}{2} .

This implies

⌊N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}−1}{2}⌉ .

Since in the current case N_{i} + B_{i} < 1 ≤ J_{1,i} + B_{i}, we obtain that N_{i} < J_{1,i}. Hence

N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < −J_{2,i} − ⋯ − J_{k−1,i} − S_{i} ≤ −\frac{m_{i}−1}{2},

and therefore

⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}}{2} ⌉ .

If one uses these findings in (2.13), the left-hand side can be bounded from below by

−m_{i} + ⌈\frac{m_{i}−1}{2}⌉ + ⌈\frac{m_{i}}{2} ⌉ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ = 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ ,

as we claimed.

Now let N_{i} + B_{i} ≥ 1. Then

N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < \frac{3}{2} − \frac{m_{i}}{2} ,

which implies

⌊N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ 1 − ⌈\frac{m_{i}}{2} ⌉ .

On the other hand, we have

N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < 1 − \frac{m_{i}}{2} ,

and hence

⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}−1}{2}⌉ .

If one uses these findings in (2.13), the left-hand side can be bounded from below by

1 − m_{i} − 1 + ⌈\frac{m_{i}}{2} ⌉ + ⌈\frac{m_{i}−1}{2}⌉ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ = 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ ,

again as we claimed.

(2) Let p < max_{1≤i≤s} 2(b_{i} − a_{i}). As in Case (1), if b_{i} − a_{i} < 0 for some i, then, since τ_{i} ≤ k − 1, we have again (2.14) and (2.15), and then the subsequent arguments show that (2.13) holds.

If b_{i} − a_{i} > 0, consider an ℓ with ℓ ≥ log_{p}(b_{i} − a_{i}) + 1. We then have ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ = 0 and

B_{i} = \{\frac{b_{i} − a_{i}}{p^{ℓ}} \} = \frac{b_{i} − a_{i}}{p^{ℓ}} ≤ \frac{b_{i} − a_{i}}{(b_{i} − a_{i})p} = \frac{1}{p} ≤ \frac{1}{2}.

Again, it should be observed that the same conclusions also hold for b_{i} − a_{i} = 0, even regardless of the value of ℓ. The arguments of Case (1) then show that (2.13) holds again.

On the other hand, if ℓ < log_{p}(b_{i} − a_{i}) + 1, then there is no restriction on B_{i} (except the trivial strict upper bound of 1). We claim that the relevant summand in (2.12) (in particular, with τ_{i} = 0) satisfies the estimation

⌊N_{i} + A_{i} + B_{i}⌋ + ⌊N_{i} + A_{i} + \frac{1}{p^{ℓ}} ⌋ − ∑_{t=1}^{k−1} ⌊J_{t,i} + B_{i}⌋

− ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ − ⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} + B_{i} − S_{i}⌋
+ 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋

≥ 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ − 1. (2.16)

Using the same notation as before, if m_{i} = 0 then one sees that (2.16) holds.

If N_{i} + B_{i} < 1, then we have

N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < 1 − \frac{m_{i}}{2} .

This implies

⌊N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}−1}{2}⌉ .

Since in the current case N_{i} + B_{i} < 1 ≤ J_{1,i} + B_{i}, we obtain that N_{i} < J_{1,i}. Hence

N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < −J_{2,i} − ⋯ − J_{k−1,i} − S_{i} ≤ −\frac{m_{i}−1}{2},

and therefore

⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}}{2} ⌉ .

If one uses these findings in (2.16), the left-hand side can be bounded from below by

− m_{i} + ⌈\frac{m_{i}−1}{2}⌉ + ⌈\frac{m_{i}}{2} ⌉ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋

= 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ ,

which is even slightly better than (2.16).

If N_{i} + B_{i} ≥ 1, then

N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < 2 − \frac{m_{i}}{2} ,

which implies

⌊N_{i} + B_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ 1 − ⌈\frac{m_{i}−1}{2}⌉ .

Furthermore, we have

N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i} < 1 − \frac{m_{i}}{2} ,

and hence

⌊N_{i} − J_{1,i} − ⋯ − J_{k−1,i} − S_{i}⌋ ≤ − ⌈\frac{m_{i}−1}{2}⌉ .

If one uses these findings in (2.16), the left-hand side can be bounded from below by

1 − m_{i} − 1 + 2 ⌈\frac{m_{i}−1}{2}⌉ + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋

≥ −1 + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋ ,

as we claimed.

If we use all this in (2.11) for fixed i (with b_{i} − a_{i} > 0), we obtain

v_{p} (\frac{(a_{i}n + a_{i})! (a_{i}n + b_{i} − 1)!}{(a_{i}j_{τ_{i}+1})! (a_{i}j_{τ_{i}+1} + b_{i} − a_{i})! ⋯ (a_{i}j_{k})! (a_{i}j_{k} + b_{i} − a_{i})!})

≥ ∑_{ℓ=1}^{⌈log_{p}(b_{i}−a_{i})⌉} (−1 + 2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ − (k − 1) ⌊\frac{b_{i}−a_{i}}{p^{ℓ}} ⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋)

+ ∑_{ℓ=⌈log_{p}(b_{i}−a_{i})⌉+1}^{∞} ∑_{i=1}^{s} (2 ⌊\frac{a_{i}−1}{p^{ℓ}}⌋ + 2 ⌊\frac{a_{i}σ}{p^{ℓ}}⌋)

≥ − ⌈log_{p}(b_{i} − a_{i})⌉ − (k − 1)v_{p}(max{b_{i} − a_{i}, 0}!) + 2v_{p}((a_{i} − 1)! (a_{i}σ)!).

Upon summation of both sides over 1 ≤ i ≤ s, this yields the claimed result in Case (2). This completes the proof of the proposition, and thus of Theorem 2. □

3. Examples

In this section, we present various examples of classical DA functions to which our results apply; many more examples of algebraic differential equations can be found in [8], for instance “Abel’s equation” on page 199 that we do not treat here. Due to the general nature of Theorems 1 and 2, the divisibility properties they imply in such or such situation are not always sharp. Better divisibility properties have often been obtained by different means, in particular when the Taylor coefficients are also known to satisfy combinatorial properties from which alternative formulas equivalent to those of type (1.5) can be deduced.

3.1. The linear case. The material in this subsection is well-known. We recall it for the reader’s convenience and also because it provides a simple interpretation of the condition that M is split over ℚ in terms of an indicial polynomial of a linear differential equation. In the general non-linear case, the interpretation of M along a similar line is given in §2.1. The situation we consider here is when Q in (1.1) is a linear polynomial in Y_{0}, …, Y_{k}, i.e., an inhomogeneous linear equation for f. Differentiating enough time, we then obtain a homogeneous differential equation for f.

A linear differential equation L : ∑_{k=0}^{n} p_{k}(x)f^{(k)} = 0, p_{0}, …, p_{n} ∈ 𝕂[X], p_{n} ≠ 0 and 𝕂 any subfield of ℂ, can be rewritten as ∑_{j=0}^{m} x^{m−j}q_{j}(θ)(f) = 0, where q_{0}, …, q_{m} ∈ 𝕂[X], q_{m} ≠ 0, and θ is Euler’s operator x\frac{d}{dx}. The polynomial q_{m}(X) is the indicial polynomial at the origin of L; see [5, §4.1, Lemma 1], the proof of which holds in fact verbatim in this general setting. Let us assume that f(x) = ∑_{n=0}^{∞} f_{n}x^{n} ∈ 𝕂[[x]] is a solution of L. It is immediate that (see [2, §4.1])

∑_{j=0}^{m} q_{j}(n + j)f_{n+j} = 0

for all n large enough, say n ≥ N. Up to increasing the value of N, we can assume that for all n ≥ N we have q_{m}(n + m) ≠ 0. Then

f_{n+m} = −\frac{1}{q_{m}(n + m)} ∑_{j=0}^{m−1} q_{j}(n + j)f_{n+j}, n ≥ N.

Let us now assume that 𝕂 = \overline{ℚ}. We may rewrite the above equation as

f_{n+1} := \frac{1}{a_{m}(n)} ∑_{j=0}^{m−1} a_{j}(n)f_{n−j}, n ≥ N + m − 1,

where a_{0}, …, a_{m} ∈ O\overline{ℚ}[X] and the leading coefficient of a_{m} is a positive integer. This recurrence is of the type in (1.5) with k_{0} = 1. By induction on n, we obtain that

D_{n} := D ∏_{j=N+m−1}^{n−1} a_{m}(j)

is a denominator of f_{n} for all n ≥ N + m, where D is a common denominator of f_{N}, …, f_{N+m−1}.

Now, a_{m} is split over ℚ if and only if the indicial polynomial at the origin q_{m} is split over ℚ. Assuming this is the case, we have a_{m}(X) = C ∏_{i=1}^{d}(u_{i}X + v_{i}) with C a nonnegative integer, u_{1}, …, u_{d} positive integers and v_{1}, …, v_{d} integers. It follows that D_{n} divides DC^{n+1}(νn + ν)!^{d} for some positive integer ν.

3.2. Roots of power series. Let f(x) = ∑_{n=0}^{∞} f_{n}x^{n} ∈ \overline{ℚ}[[x]] be a solution of a linear equation with coefficients in \overline{ℚ}[x] and with rational exponents at the origin. We have seen in §3.1 that the sequence (f_{n})_{n≥0} is then solution of a recurrence of type (1.5), which in this case reads

∑_{k=0}^{d} p_{k}(n)f_{n+k} = 0,  (3.1)

with p_{k}(X) ∈ O\overline{ℚ}[X] for k = 0, …, d − 1, and p_{d}(X) ∈ ℚ[X] and split over ℚ.

For simplicity, let us assume without loss of generality that f_{0} ≠ 0. Let m ≥ 2 be an integer and let g(x) = ∑_{n=0}^{∞} g_{n}x^{n} ∈ \overline{ℚ}[X] be such that f = g^{m}. In particular, g_{0} ≠ 0 because f_{0} = g_{0}^{m}. Substituting g^{m} for f in the differential equation satisfied by f readily implies that g is a DA function. We claim that the sequence (g_{n})_{n≥0} is also solution of a recurrence of type (1.5).

Indeed, we have

f_{n} = ∑_{j_{1}+j_{2}+···+j_{m}=n} g_{j_{1}}g_{j_{2}} ⋯ g_{j_{m}} = mg_{0}^{m−1}g_{n} + ∑_{j_{1}+j_{2}+···+j_{m}=n_{j_{1},...,jm≤n−1}} g_{j_{1}}g_{j_{2}} ⋯ g_{j_{m}},

so that from (3.1) we deduce that

mg_{0}^{m−1}p_{d}(n)g_{n+d}

= − ∑_{k=0}^{d−1} (p_{k}(n) ∑_{j_{1}+j_{2}+···+j_{m}=n+k} g_{j_{1}}g_{j_{2}} ⋯ g_{j_{m}}) − p_{d}(n) ∑_{j_{1}+j_{2}+···+j_{m}=n+d_{j_{1},...,jm≤n+d−1}} g_{j_{1}}g_{j_{2}} ⋯ g_{j_{m}}. (3.2)

Since p_{d}(n) ≠ 0 for all n large enough, it follows that (3.2) can be rearranged into a recurrence of type (1.5) as claimed.

3.3. Generating functions of special sequences. We now illustrate our method with three classical series solutions of Riccati equations.

The function x/ log(1 + x) = ∑_{n=0}^{∞} g_{n}x^{n} is solution of x(1 + x)y′ + y^{2} − (1 + x)y = 0. This differential equation provides the recurrence relation

g_{n+1} = −\frac{1}{n + 2}((n − 1)g_{n} + ∑_{j=1}^{n} g_{j}g_{n+1−j}), n ≥ 0,  (3.3)

with g_{0} = 1. The g_{n} are known as the Gregory coefficients or Cauchy numbers of the first kind. It is known that n! g_{n} = ∑_{k=0}^{n} s(n, k)/(k +1), where s(n, k) ∈ ℤ are Stirling numbers of the first kind, so that lcm{1, 2, …, n + 1}n! g_{n} ∈ ℤ for all n ≥ 0. See [4, pp. 293–294].

Let us explain what divisibility could be obtained with our result. The recurrence (3.3) is an example for which σ_{1} = −1, and thus this necessitates to transform it into a recurrence with σ ≥ 0. Following the procedure in the proof of Lemma 1, let g̃_{n} = g_{n+1}, substitute this in (3.3), and finally replace n by n + 1. In this manner, we obtain

g̃_{n+1} = −\frac{1}{n + 3}(ng̃_{n} + ∑_{j=0}^{n} g̃_{j}g̃_{n−j}), n ≥ 0, g̃_{0} = 1/2.

We can apply Theorem 2 with s = 1, a_{1} = 1, b_{1} = 3, C = 1, D = 2, σ_{1} = σ_{2} = 0, k_{0} = 2, N = 0 and Π = 12, so that 24^{n}(n − 1)! (n + 1)! g_{n} ∈ ℤ for all n ≥ 1.

Next we consider the function \frac{x}{e^{x}−1} = ∑_{n=0}^{∞} b_{n}x^{n}. We have b_{0} = 0, b_{1} = −1/2, b_{2n+1} = 0 for n ≥ 2. Let us write b_{2n} = B_{2n}/(2n)! for n ≥ 1. The numbers B_{2n} are the Bernoulli numbers. The Clausen–von Staudt Theorem says that the denominator of B_{2n} is the product of the prime numbers p such that p−1 divides 2n; this quantity divides lcm{1, 2, …, 2n+1}.

The above power series is solution of xy′ + y^{2} + (x − 1)y = 0, from which we deduce that

b_{n+1} = −\frac{1}{n + 2}(b_{n} + ∑_{j=1}^{n} b_{j}b_{n+1−j}), n ≥ 0, b_{0} = 1.  (3.4)

This is again an example of a recurrence with σ_{1} = −1, and it is completely similar to (3.3). We let b̃_{n} := b_{n+1}, substitute in (3.4), and then change n to n + 1. We obtain

b̃_{n+1} = −\frac{1}{n + 3}(b̃_{n} + ∑_{j=0}^{n} b̃_{j}b̃_{n−j}), n ≥ 0, b̃_{0} = 1/2.

Exactly as with the Gregory coefficients, we can apply Theorem 2 with s = 1, a_{1} = 1, b_{1} = 3, C = 1, D = 2, σ_{1} = σ_{2} = 0, k_{0} = 2, N = 0 and Π = 12, so that 24^{n}(n−1)! (n+1)! b_{n} ∈ ℤ for all n ≥ 1.

The function tan(x) = ∑_{n=0}^{∞} t_{n}x^{n} is solution of y′ − y^{2} − 1 = 0, which translates into the recurrence

t_{n+1} = \frac{1}{n + 1} ∑_{j=0}^{n} t_{j}t_{n−j}, n ≥ 1,

with t_{0} = 0, t_{1} = 1. We can apply Theorem 2 with C = 1, D = 1, s = 1, a_{1} = 1, b_{1} = 1, σ_{1} = σ_{2} = 0, k_{0} = 2, P_{0,1} = 0, P_{0,2} = 1, N = 1 and Π = 1. It follows that n!^{2}t_{n} ∈ ℤ, n ≥ 0. It is known that t_{2n} = 0 while t_{2n−1} = (−1)^{n−1}4^{n}(4^{n} − 1)B_{2n}/(2n)!, where the B_{2n} are again the Bernoulli numbers. See [4, p. 88].

These examples show why Theorem 2 is not always sharp: quantities like lcm{1, 2, …, n} or ∏_{p: p−1|n} p are “incorporated” into an n! factor, because even though they are less than 3^{n}, they do not divide δ^{n} for some δ. At the cost of a complicated p-adic analysis that we do not reproduce here, we have been able to prove that for the sequence (f_{n})_{n≥0} defined in the introduction by (1.3) with a = 1, the denominator of f_{n} divides (n + 1)!^{2}/ lcm{1, 2, …, n + 1}, which improves on n! (n + 1)! given by Theorem 2. We did not try to prove such refinements in the general case.

3.4. Elliptic differential equation. The Weierstraß elliptic function ℘ with modular invariants g_{2}, g_{3} ∈ ℂ (such that g_{2}^{3} ≠ 27g_{3}^{2}) satisfies the algebraic differential equation

℘^{′2} = 4℘^{3} − g_{2}℘ − g_{3}.  (3.5)

The function ℘(x) is a doubly-periodic meromorphic function on ℂ, and it can be expanded in a Laurent series at 0 (^{3}). We have

℘(x) = \frac{1}{x^{2}} + ∑_{n=2}^{∞} p_{n}x^{2n−2},

where the coefficients p_{n} satisfy the recurrence

p_{n} = \frac{3}{(2n + 1)(n − 3)} ∑_{j=2}^{n−2} p_{j}p_{n−j}, n ≥ 4,  (3.6)

with p_{2} = g_{2}/20, p_{3} = g_{3}/28; see [1, p. 635, Eq. (18.5.3)]. Eq. (3.6) is easily obtained not from (3.5) but from the simpler equation

12℘^{2} − 2℘′′ − g_{2} = 0,

which follows at once from differentiating (3.5). To apply Theorem 2, we define u_{n} := p_{n+2} for n ≥ 0: we have

u_{n+1} = \frac{3}{(2n + 7)n} ∑_{j=0}^{n−1} u_{j}u_{n−j−1}, n ≥ 1,

with u_{0} = g_{2}/20 and u_{1} = g_{3}/28. It is obvious that u_{n} is a polynomial in g_{2} and g_{3} with rational coefficients. Assuming that g_{2}, g_{3} are in \overline{ℚ}, let D denote a common denominator of g_{2}/20 and g_{3}/28. We are in the situation of Theorem 2 with C = 1, s = 2, a_{1} = 1, b_{1} = 0, a_{2} = 2, b_{2} = 7, k_{0} = 2, σ_{2} = 1, N = 1, P_{0,1} = P_{0,2} = P_{1,0} = 0 and P_{1,1} = 3 and Π := 2^{3} · 3^{2} · 5 · 7 = 2520. It follows that

D^{n+1}Π^{n}(n − 1)! n! (2n)! (2n + 5)! p_{n+2} ∈ O\overline{ℚ}, n ≥ 1.

We have not been able to find a divisibility of this nature in this generality in the literature, but would be surprised if it were a new result. For instance, Hurwitz obtained a much better result when g_{2} = 4 and g_{3} = 0 (the lemniscate case). Using his notations, let

℘(x) = \frac{1}{x^{2}} + ∑_{n=1}^{∞} \frac{2^{4n}E_{n}}{4n} · \frac{x^{4n−2}}{(4n − 2)!} ∈ ℚ[[x]],

which is a solution of y^{′2} = 4y^{3} − 4y. Then he showed in [7, p. 208] the following result, which is an analogue of the Clausen–von Staudt Theorem for the Bernoulli numbers.

Let d_{n} denote the denominator of E_{n}: (1) there is no prime number ≡ 3 mod 4 that divides d_{n}; (2) if a prime number p ≡ 1 mod 4 divides d_{n}, then p − 1 divides 4n and p^{2} does not divide d_{n}.

3Our results hold for DA Laurent series f, because x^{m}f is a DA power series for some suitable integer m.

A non-linear recurrence of type (1.4) for the coefficients of x^{m}f obviously provides one for those of f.

Moreover, it follows from the partial fraction expansion of E_{n} in [7, p. 219, Eq. (78)] that v_{2}(E_{n}) = −1. Hence, (∏_{p: p−1|4n} p)E_{n} ∈ ℤ. Moreover, letting

\frac{1}{2℘(x)} = ∑_{n=0}^{∞} e_{n} \frac{x^{4n+2}}{(4n + 2)!},

he also proved that e_{n} ∈ ℤ for all n ≥ 0; see [7, p. 224]. Finally, let us mention that −1/℘(x) is curiously also a solution of y^{′2} = 4y^{3} − 4y, and that \sqrt{1/℘(x)} is a solution of y′′ + 2y^{3} = 0.

3.5. Painlevé equations. Painlevé equations produce a lot of interesting examples. (We refer the reader to [3] for an introduction.) In this subsection, we only consider the family of algebraic differential equations known as Painlevé PII’:

y′′ = δ(2y^{3} − 2xy) + γ(6y^{2} + x) + βy + α,

where it is assumed that α, β, γ, δ are in \overline{ℚ}. It is a simple task to check that u(x) := ∑_{n=0}^{∞} u_{n}x^{n} is a solution of PII’, where the sequence (u_{n})_{n≥0} is given by

u_{n+1} = \frac{1}{n(n + 1)}(2δ ∑_{j_{1}+j_{2}+j_{3}=n−1_{0≤j_{1},j_{2},j_{3}≤n−1}} u_{j_{1}}u_{j_{2}}u_{j_{3}} + 6γ ∑_{j_{1}+j_{2}=n−1_{0≤j_{1},j_{2}≤n−1}} u_{j_{1}}u_{j_{2}} − 2βu_{n−1} − 2δu_{n−2}), n ≥ 2,

with u_{0}, u_{1} arbitrary in \overline{ℚ}, and u_{2} = −δu_{0}^{3} − 3γu_{0}^{2} − βu_{0}/2 − α/2. This is a situation where Theorem 2 with s = 2, a_{1} = a_{2} = b_{1} = 1, b_{2} = 0, k_{0} = 3, σ_{1} = 1, σ_{2} = 2, N = 2, Π = 1, and C the common denominator of 2δ, 6γ, 2β, applies. Letting D ≥ 1 denote the least common denominator of u_{0}, u_{1}, u_{2}, we have

C^{n}D^{2n+1}(n − 1)! n!^{3} u_{n} ∈ ℤ, n ≥ 2.

3.6. Kepler’s equation. Kepler’s equation from Celestial Mechanics is E − e sin(E) = M, where E, M, e are the eccentric anomaly, the mean anomaly, and the eccentricity, respectively. Inventing for the purpose of solving this equation what is now known as Lagrange’s inversion formula, Lagrange [11] found the well-known expansion

φ(e) := E − M = ∑_{n=1}^{∞} (\frac{d}{dt})^{n−1}( sin(t)^{n})_{t=M} · \frac{e^{n}}{n!} ,  (3.7)

which is an odd and 2π-periodic function of M, and which converges in the disk |e| < e_{0} for all M. Here, e_{0} = 2\sqrt{ρ(1 − ρ)}/(1 − 2ρ) ≈ 0.6627, where ρ ≈ 0.0832 satisfies 1/ρ − 1 = exp(2/(1 − 2ρ)); see the details in [21].

It is easy to prove that (sin^{n}(t))^{(n−1)} = P_{n}( sin(t), cos(t)), where P_{n}(X, Y) ∈ ℤ[X, Y] is of degree ≤ n in X and ≤ 1 in Y. Therefore, when sin(M) ∈ \overline{ℚ} (hence cos(M) ∈ \overline{ℚ} as well), there exists a positive integer δ such that, for all n ≥ 1,

δ^{n+1}(\frac{d}{dt})^{n−1}( sin^{n}(t))_{t=M} ∈ O\overline{ℚ}.

Hence, from (3.7), we deduce in this case that the denominator of the n-th Taylor coefficient of φ at the origin divides δ^{n+1}n!.

We now want to use this example to show how to effectively determine a recurrence like (1.5) in a more complex situation than in the previous examples. We change e to x and let φ(x) = E − M. Kepler’s equation reads sin(φ + M) = φ/x, with φ(0) = 0 and φ′(0) = sin(M). Differentiating this equation with respect to x and squaring both sides, we find

φ^{′2}(1 − (φ/x)^{2}) = ((φ/x)′)^{2},  (3.8)

or, in expanded form,

x^{2}(x^{2} − 1)φ^{′2} − φ^{2} + 2xφ′φ − x^{2}φ^{′2}φ^{2} = 0.  (3.9)

Let φ(x) = ∑_{n=0}^{∞} ϕ_{n}x^{n}. We have ϕ_{0} = 0, ϕ_{1} = sin(M), and ϕ_{2} = sin(M) cos(M). We shall assume below that ϕ_{2} ≠ 0, which a fortiori implies that ϕ_{1} ≠ 0. From the algebraic differential equation (3.9), we deduce that, for n ≥ 5, we have

∑_{j_{1}+j_{2}=n−4_{0≤j_{1},j_{2}≤n−4}}(j_{1} + 1)(j_{2} + 1)ϕ_{j_{1}+1}ϕ_{j_{2}+1} − ∑_{j_{1}+j_{2}=n−2_{0≤j_{1},j_{2}≤n−2}}(j_{1} + 1)(j_{2} + 1)ϕ_{j_{1}+1}ϕ_{j_{2}+1} − ∑_{j_{1}+j_{2}=n_{0≤j_{1},j_{2}≤n}} ϕ_{j_{1}}ϕ_{j_{2}}

+ ∑_{j_{1}+j_{2}=n−1_{0≤j_{1},j_{2}≤n−1}} 2(j_{1} + 1)ϕ_{j_{1}+1}ϕ_{j_{2}} − ∑_{j_{1}+j_{2}+j_{3}+j_{4}=n−2_{0≤j_{1},j_{2},j_{3},j_{4}≤n−2}}(j_{1} + 1)(j_{2} + 1)ϕ_{j_{1}+1}ϕ_{j_{2}+1}ϕ_{j_{3}}ϕ_{j_{4}} = 0,

which can be rewritten as

∑_{j_{1}+j_{2}=n−2_{0≤j_{1},j_{2}≤n−3}} j_{1}j_{2}ϕ_{j_{1}}ϕ_{j_{2}} − ∑_{j_{1}+j_{2}=n_{0≤j_{1},j_{2}≤n−1}} j_{1}j_{2}ϕ_{j_{1}}ϕ_{j_{2}} − ∑_{j_{1}+j_{2}=n_{0≤j_{1},j_{2}≤n}} ϕ_{j_{1}}ϕ_{j_{2}}

+ ∑_{j_{1}+j_{2}=n_{0≤j_{1}≤n,0≤j_{2}≤n−1}} 2j_{1}ϕ_{j_{1}}ϕ_{j_{2}} − ∑_{j_{1}+j_{2}+j_{3}+j_{4}=n_{0≤j_{1},j_{2}≤n−1,0≤j_{3},j_{4}≤n−2}} j_{1}j_{2}ϕ_{j_{1}}ϕ_{j_{2}}ϕ_{j_{3}}ϕ_{j_{4}} = 0.

This is equivalent to

∑_{j_{1}+j_{2}=n−2_{0≤j_{1},j_{2}≤n}} P_{1}(n, j_{1}, j_{2})ϕ_{j_{1}}ϕ_{j_{2}} + ∑_{j_{1}+j_{2}=n_{0≤j_{1},j_{2}≤n}} P_{2}(n, j_{1}, j_{2})ϕ_{j_{1}}ϕ_{j_{2}}

+ ∑_{j_{1}+j_{2}+j_{3}+j_{4}=n_{0≤j_{1},j_{2},j_{3},j_{4}≤n}} P_{3}(n, j_{1}, j_{2}, j_{3}, j_{4})ϕ_{j_{1}}ϕ_{j_{2}}ϕ_{j_{3}}ϕ_{j_{4}} = 0,

where

P_{1}(n, j_{1}, j_{2}) = j_{1}j_{2}1_{{0≤j_{1},j_{2}≤n−3}},
P_{2}(n, j_{1}, j_{2}) = −j_{1}j_{2}1_{{0≤j_{1},j_{2}≤n−1}} + 2j_{1}1_{{0≤j_{2}≤n−1}} − 1,
P_{3}(n, j_{1}, j_{2}, j_{3}, j_{4}) = −j_{1}j_{2}1_{{0≤j_{1},j_{2}≤n−1,0≤j_{3},j_{4}≤n−2}}.

This is an algebraic relation between ϕ_{0}, ϕ_{1}, …, ϕ_{n}. A simple analysis shows that neither ϕ_{n} nor ϕ_{n−1} appear in this relation. Indeed, in the three sums, ϕ_{n} can appear only in terms involving ϕ_{n}ϕ_{0} or ϕ_{n}ϕ_{0}^{3}, which are equal to 0 because ϕ_{0} = 0. The coefficient ϕ_{n−1} does not appear in the first sum, appears in the second as (P_{2}(n, n−1, 1)+P_{2}(n, 1, n−1))ϕ_{1}ϕ_{n−1} = 0 because P_{2}(n, n − 1, 1) + P_{2}(n, 1, n − 1) = 0, and in the third sum in terms of the form ϕ_{n−1}ϕ_{1}ϕ_{0}ϕ_{0} = 0. The coefficient ϕ_{n−2} does not appear in the first sum, appears in terms involving ϕ_{n−2}ϕ_{1}^{2}ϕ_{0} = 0 and ϕ_{n−2}ϕ_{2}ϕ_{0}^{2} = 0 in the third sum, and in the second sum, ϕ_{n−2} appears in the expression (P_{2}(n, n − 2, 2) + P_{2}(n, 2, n − 2))ϕ_{2}ϕ_{n−2} = −2(n + 1)ϕ_{2}ϕ_{n−2}. Therefore, since we have assumed that ϕ_{2} ≠ 0, we finally obtain the recurrence (after changing n to n + 3)

ϕ_{n+1} = \frac{1}{2ϕ_{2}(n + 4)}( ∑_{j_{1}+j_{2}=n+1_{0≤j_{1},j_{2}≤n}} P_{1}(n + 3, j_{1}, j_{2})ϕ_{j_{1}}ϕ_{j_{2}} + ∑_{j_{1}+j_{2}=n+3_{0≤j_{1},j_{2}≤n}} P_{2}(n + 3, j_{1}, j_{2})ϕ_{j_{1}}ϕ_{j_{2}}

+ ∑_{j_{1}+j_{2}+j_{3}+j_{4}=n+3_{0≤j_{1},j_{2},j_{3},j_{4}≤n}} P_{3}(n + 3, j_{1}, j_{2}, j_{3}, j_{4})ϕ_{j_{1}}ϕ_{j_{2}}ϕ_{j_{3}}ϕ_{j_{4}}), n ≥ 2.

This is a recurrence of the form (1.5) (with negative σ_{1}), to which we can apply Theorem 1 because M(X) ∈ ℚ[X] is linear. We do not provide the details of the explicit result, which is of the form δ̃^{n+1}(νn + ν)!^{2} for some integers δ̃, ν ≥ 1, because it is weaker than the already obtained divisibility of δ^{n+1}n! by the denominator of ϕ_{n}.

Finally, instead of φ, we could consider ψ(x) := φ(x)/x = ∑_{n=0}^{∞} ϕ_{n+1}x^{n}. The differential equation (3.8) becomes (xψ)^{′2}(1 − ψ^{2}) = (ψ′)^{2} or, in expanded form,

(1 − x)ψ^{′2} + 2xψ′ψ^{3} + x^{2}ψ^{′2}ψ^{2} − 2xψ′ψ + ψ^{4} − ψ^{2} = 0.

This would lead to an a priori different recurrence for the sequence (ϕ_{n})_{n≥0}.

3.7. The compositional inverse of the dilogarithm. The dilogarithm is Li_{2}(x) := ∑_{n=1}^{∞} x^{n}/n^{2}, with derivative − log(1 − x)/x, and it admits a power series ℓ as inverse for the composition:

ℓ(x) := ∑_{n=0}^{∞} ℓ_{n}x^{n} = x − \frac{1}{4}x^{2} + \frac{1}{72}x^{3} − \frac{1}{576}x^{4} − \frac{31}{86400}x^{5} − \frac{149}{1036800}x^{6} − · · · .

The function ℓ is DA, for it satisfies

ℓ′′ℓ − ℓ′′ℓ^{2} + ℓ^{′3} + ℓ^{′2}ℓ − ℓ^{′2} = 0.  (3.10)

Indeed, we have 1 = (Li_{2}(ℓ))′ = −ℓ′ log(1 − ℓ)/ℓ, so that, by multiplying by ℓ and then differentiating both sides, we obtain

ℓ′ = −ℓ′′ log(1 − ℓ) + \frac{ℓ^{′2}}{1 − ℓ} = \frac{ℓ′′ℓ}{ℓ′} + \frac{ℓ^{′2}}{1 − ℓ},

which gives the above differential equation after some rearrangement. The situation is a little simpler than with Kepler’s equation because (3.10) is autonomous, i.e., its coefficients are independent of x. For all n ≥ 0, we have

[x^{n}](ℓ′′ℓ) = ∑_{i+j=n}(i + 1)(i + 2)ℓ_{i+2}ℓ_{j},  (3.11)
[x^{n}](−ℓ′′ℓ^{2}) = − ∑_{i+j+k=n}(i + 1)(i + 2)ℓ_{i+2}ℓ_{j}f_{k},
[x^{n}](ℓ^{′3}) = ∑_{i+j+k=n}(i + 1)(j + 1)(k + 1)ℓ_{i+1}ℓ_{j+1}ℓ_{k+1},  (3.12)
[x^{n}](ℓ^{′2}ℓ) = ∑_{i+j+k=n}(i + 1)(j + 1)ℓ_{i+1}ℓ_{j+1}ℓ_{k},
[x^{n}](−ℓ^{′2}) = − ∑_{i+j=n}(i + 1)(j + 1)ℓ_{i+1}ℓ_{j+1},  (3.13)

where in the summations all the indices run between 0 and n, and where as usual [x^{n}](f) denotes the n-th Taylor coefficient of a power series f. Proceeding as in §3.6, we see that ℓ_{n+2} appears in none of these sums because it is always multiplied by ℓ_{0} = 0. The term ℓ_{n+1} appears only in (3.11), (3.12), and in (3.13), with the coefficients n(n + 1)ℓ_{1}, 3(n + 1)ℓ_{1}^{2}, and −2(n + 1)ℓ_{1}, respectively. This is ℓ_{1}(n + 1)(n + 3ℓ_{1} − 2) = (n + 1)^{2} in total. Therefore, we have

ℓ_{n+1} = \frac{1}{(n + 1)^{2}} ( − ∑_{i+j=n_{i≤n−2,j≤n}}(i + 1)(i + 2)ℓ_{i+2}ℓ_{j} + ∑_{i+j+k=n_{i≤n−2,j,k≤n}}(i + 1)(i + 2)ℓ_{i+2}ℓ_{j}ℓ_{k}

− ∑_{i+j+k=n_{i,j,k≤n−1}}(i + 1)(j + 1)(k + 1)ℓ_{i+1}ℓ_{j+1}ℓ_{k+1} − ∑_{i+j+k=n_{i,j≤n−1,k≤n}}(i + 1)(j + 1)ℓ_{i+1}ℓ_{j+1}ℓ_{k}

+ ∑_{i+j=n_{i,j≤n−1}}(i + 1)(j + 1)ℓ_{i+1}ℓ_{j+1}), n ≥ 0.

It follows that, for all n ≥ 0, δ^{n+1}(νn+ν)!^{4}ℓ_{n} ∈ ℤ for some positive integers δ, ν. Numerical experiments suggest that the exponent 4 could be replaced by 2. This would be true if it could be proved for instance that (ℓ_{n})_{n≥0} satisfies a non-linear recurrence of type (1.5) like above but with (n + 1)^{2} replaced by an + b for suitable integers a and b.

As a side remark, we point out that it is possible to determine the asymptotic behaviour of ℓ_{n}. By Lagrange’s inversion formula [6, p. 732, Theorem A.2], we have

ℓ_{n} = \frac{1}{2iπn} ∫_{C} \frac{dz}{Li_{2}(z)^{n}} , n ≥ 0,

where C is a circle centred at 0 and of sufficiently small radius. This circle can be deformed to a suitable Hankel-type contour to which the saddle point method can be applied (see [6] for similar computations). We obtain that

ℓ_{n} ∼ −\frac{ζ(2)^{−n}}{n^{2} log(n)^{2}} , n → +∞.  (3.14)

Hence, the radius of convergence of ℓ at x = 0 is π^{2}/6, and (ℓ_{n}/ℓ_{n+1})_{n≥0} is a sequence of rational numbers that converge to π^{2}/6, but not especially quickly. The proof of (3.14) is given in our note [10].

3.8. Examples related to modular forms. Various automorphic functions, including certain classical modular forms, are known to satisfy algebraic differential equations of order at least 3 and of the form Q(y)(y′)^{2} = \{y; x\}, where \{y; x\} := y′′′/y′ − \frac{3}{2}(y′′/y′)^{2} is the Schwarzian derivative with respect to x, and Q is a rational function with poles of order at most 2. For instance, Mahler proved in [13] that the elliptic modular invariant

F (q) := J(q^{2}) = q^{−2}(1 + 240 ∑_{n=1}^{∞} n^{3}q^{2n}(1 − q^{2n})^{−1})^{3} ∏_{n=1}^{∞}(1 − q^{2n})^{−24}

= 1/q^{2} + 744 + 196884q^{2} + 21493760q^{4} + 864299970q^{6} + ⋯,

satisfies the order 3 equation

F′′′ = \frac{3q^{2}F^{′′2} − 4qF′F′′ − F^{′2}}{2q^{2}F′} − F^{′3}(\frac{4}{9F^{2}} + \frac{3}{8(F − 12^{3})^{2}} − \frac{23}{72F (F − 12^{3})}), (3.15)

where the derivatives are taken with respect to q. He also proved that F cannot satisfy an algebraic differential equation of order ≤ 2. Since the degree in F′′′ of (3.15) is 1, this equation is of the standard form to which we can apply the procedure described in [23, p. 382]. The indicial polynomial P_{0} of the associated differential operator L_{0} ∈ \overline{ℚ}[[x]][\frac{d}{dx}] is P_{0}(X) = X^{3} + 8X^{2} − 10X + 64, which has no rational root. Lemma 2.2 of [23, p. 382] then provides a non-linear recurrence of type (1.4) for the coefficients of F with M(X) = P_{0}(X+m) for some integer m ≥ 0. This recurrence is not of type (1.5), and our Theorems 1 and 2 cannot be applied to it. We do not know if there exists a recurrence of type (1.5) for the coefficients of F (which are well-known to be integers), i.e., with M split over ℚ. Let us also mention that a vast algebraico-geometric generalization of modular forms, known as mirror maps, are shown to be DA functions in [25, Proposition 1].

3.9. A simple example with M non split over ℚ. We conclude with an example for which M is non-split over ℚ, and for which it is possible to make numerical experiments. The differential equation x^{2}y′′ + (x − 1)y′ + y − xy^{2} = 1 has as solution the power series ∑_{n=0}^{∞} f_{n}x^{n} with

f_{n+1} = \frac{1}{n^{2} + 1} ∑_{k=0}^{n} f_{k}f_{n−k}, n ≥ 0, f_{0} := 1.

Numerical experiments for values of n up to 2000 suggest that the denominator d_{n} of f_{n} is such that log(d_{n})/(n log(n)^{2}) converges to a constant close to 0.566. Moreover,

\frac{log(d_{2n})}{2n log(2n)} − \frac{log(d_{n})}{n log(n)}

seems to converge to a constant close to 0.39, which tends to confirm that log(d_{n})/(n log(n)) behaves more like log(n) than like log log(n).

Moreover, it seems that ∏_{k=0}^{⌊n/2⌋}(k^{2} + 1) divides 2^{n}d_{n} for all n ≥ 0. If true, this conjecturally implies that there exist no integers δ, ν, μ, s ≥ 0 such that d_{n} divides δ^{n+1}(νn + μ)!^{s} for all n ≥ 0. Indeed, on the contrary, this would imply the same type of divisibility for ∏_{k=0}^{n}(k^{2} + 1), and we have explained in the introduction why this is not true conjecturally.

All these computations do not point towards a positive solution to Mahler’s denominator conjecture, but these experiments should be conducted for more values of n on this sequence and others in the non-split case to be more affirmative.

Acknowledgement

We are grateful to the Institut Fourier for having supported a research visit of the first author at the institute in November 2024, during which this project, started at the occasion of another research visit at the institute in 2018, could be eventually finalised.

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover, 1972.

[2] V. V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi–Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), 493-–533.

[3] P. Clarkson, Painlevé equations — nonlinear special functions, in: Orthogonal Polynomials and Special Functions. Computation and Applications, Lecture notes from the 5th European Summer School held at the Universidad Carlos III de Madrid, Leganés, July 8–18, 2004, F. Marcellán and W. Van Assche (eds.), Lecture Notes in Math., vol. 1883, Springer-Verlag, Berlin, 2006.

[4] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht, 1974.

[5] S. Fischler and T. Rivoal, Linear independence of values of G-functions, J. Eur. Math. Soc. 22.5 (2020), 1531–1576.

[6] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.

[7] A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Annal. 51 (1898), 196–226.

[8] E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, I, Gewöhnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977.

[9] C. Krattenthaler and T. Rivoal, Hypergéométrie et fonction zêta de Riemann, Mem. Amer. Math. Soc. 186 (2007), 93 pages.

[10] C. Krattenthaler and T. Rivoal, On the compositional inverse of polylogarithms, unpublished note, 2025, 6 pages.

[11] J.-L. Lagrange, Sur le problème de Kepler, Mémoires de l’Académie Royale des Sciences de Berlin 25 (1771), 204–233.

[12] A.-M. Legendre, Essai sur la théorie des nombres, 2ed., Courcier, Paris, 1808.

[13] K. Mahler, On algebraic differential equations satisfied by automorphic functions, J. Austral. Math. Soc. 10 (1969), 445–450.

[14] K. Mahler, A p-adic analogue of a theorem by J. Popken, J. Austral. Math. Soc. 16 (1973), 176–184.

[15] K. Mahler, Lectures on Transcendental Numbers, Lecture Notes in Math., vol. 546, Springer-Verlag, Berlin/New York, 1976.

[16] E. Maillet, Sur les séries divergentes et les équations différentielles, Ann. Sci. ENS (3) 20 (1903), 487–518.

[17] B. Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2.1 (1989), 1–4.

[18] O. Perron, Über lineare Differentialgleichungen mit rationalen Koeffizienten, Acta Math. 34 (1911), 139–163.

[19] G. Pólya, Über das Anwachsen von ganzen Funktionen, die einer Differentialgleichung genügen, Vierteljahrschr. Naturf. Ges. Zürich 61 (1916), 531–545.

[20] J. Popken, Über arithmetische Eigenschaften analytischer Funktionen, Dissertation: Groningen. Noord-Hollandsche Uitgeversmaatschappij, Amsterdam, 1935.

[21] V. Puiseux, Sur la convergence des séries qui se présentent dans la théorie du mouvement elliptique des planètes, J. Math. Pures Appl. 14 (1849), 33–39.

[22] A. Sah, An improved bound on the least common multiple of polynomial sequences, J. Théor. Nombres Bordeaux 32 (2020), 891–899.

[23] Y. Sibuya and S. Sperber, Some new results on power-series solutions of algebraic differential equations, in: Singular perturbations and asymptotics, Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., vol. 45, R. E. Meyer and S. V. Parter (eds.), Academic Press, New York–London, 1980, pp. 379–404.

[24] Y. Sibuya and S. Sperber, Arithmetic properties of power series solutions of algebraic differential equations, Ann. Math. (2) 113 (1981), 111–157.

[25] W. Zudilin, Transcendence problems of the mirror [Number theory casting a look at the mirror],

    preprint 2001, available at https://arxiv.org/abs/math/0008237

C. Krattenthaler, Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-

Platz 1, A-1090 Vienna, Austria. WWW: http://www.mat.univie.ac.at/~kratt.

T. Rivoal, Institut Fourier, Université Grenoble Alpes, CNRS, CS 40700, 38058 Greno-

ble cedex 9, France. WWW: https://rivoal.perso.math.cnrs.fr.