SUMMATION FORMULAS FOR GJMS-OPERATORS AND Q-CURVATURES ON THE MÖBIUS SPHERE

ANDREAS JUHL AND CHRISTIAN KRATTENTHALER

Dedicated to Richard Askey

Abstract. For the Möbius spheres 𝕊^{q,p}, we give alternative elementary proofs of the recursive formulas for GJMS-operators and Q-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278–1370]. These proofs make essential use of the theory of hypergeometric series.

February 14, 2014

Contents

1. Introduction and statement of results 1 2. Proof of Theorem 1.1 6 3. Proof of Theorem 1.2 16 Appendix. Proofs of some auxiliary results 19 References 20

1. Introduction and statement of results

In [J10] and [J13], one of the authors proved recursive formulas for GJMS-operators and Branson’s Q-curvatures. In the present article, we give alternative proofs of these results for the space 𝕊^{q,p} = 𝕊^{q} × 𝕊^{p} with the signature (q, p)-metric g_{𝕊^{q}} − g_{𝕊^{p}} given by the round metrics on the factors. The corresponding (explicit) formulas are found in Theorems 1.1 and 1.2. The proofs which we give here provide another example of the extreme usefulness of hypergeometric series identities. For other recent instances of their crucial role in conformal differential geometry we refer to [CKOP11] and [K13].

In the following, we write n = q + p. A GJMS-operator P_{2N}, N ≥ 1, is a specific rule which associates to any pseudo-

Riemannian manifold (M, g) a differential operator of the form

P_{2N}(g) = Δ_{g}^{N} + lower-order terms,

2010 Mathematics Subject Classification. Primary 53B20 53B30; Secondary 05A19 33C20 53A30.

The work of the first author was supported by SFB 647 “Raum-Zeit-Materie” of DFG. The work of the second author was 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.”

where Δ_{g} = −δ_{g}d is the Laplace–Beltrami operator of g. The operators P_{2N}(g) are given by universal polynomial formulas in terms of the metric g, its inverse, the curvature tensor of g, and its covariant derivatives. Moreover, the operators P_{2N}(g) are conformally covariant in the sense that

e^{( \frac{n}{2} +N)φ}P_{2N}(e^{2φ}g)(u) = P_{2N}(g) (e^{( \frac{n}{2} −N)φ}u)  (1.1)

for all φ ∈ C^{∞}(M) and u ∈ C^{∞}(M). The operators P_{2N}(g) were derived in [GJMS92] from the N^{th} powers of the Laplace–Beltrami operator of the Fefferman–Graham ambient metric associated to g [FG12a]. On manifolds of even dimension n, this construction is obstructed at 2N = n, i.e., it only yields a finite sequence P_{2}(g), …, P_{n}(g) of operators for which (1.1) is valid for arbitrary metrics. However, the obstructions should not be attributed to the method of construction. In fact, Graham [G92] proved that on manifolds of dimension 4, there is no conformally covariant cube of the Laplacian. More generally, Gover and Hirachi [GH04] proved that on manifolds of even dimension n, there is no conformally covariant power of the Laplacian the order of which exceeds n. Nevertheless, for even n and locally conformally flat metrics, the obstruction vanishes, and the construction of [GJMS92] still yields an infinite sequence of conformally covariant operators. In the present work, we will be concerned with such a case. For odd n, the operators P_{2N}(g) exist for general metrics g and all even orders 2N ≥ 2.

The GJMS-operators are not uniquely determined by the requirement of their conformal covariance. It remains a challenge to characterize them among other conformally covariant operators with leading part a power of the Laplacian.

Explicit formulas for P_{2} and P_{4} are well-known. These operators coincide with the Yamabe-operator

Δ − (\frac{n}{2} − 1) J, J = \frac{scal}{2(n − 1)},  (1.2)

and the Paneitz-operator

Δ^{2} + δ ((n − 2)Jg − 4P) d + (\frac{n}{2} − 2) (\frac{n}{2}J^{2} − 2P^{2} − ΔJ) .  (1.3)

Here

P = \frac{1}{n − 2} (Ric −Jg)

denotes the Schouten tensor; in (1.3), it is regarded as an endomorphism on one-forms.

Explicit formulas for higher-order GJMS-operators are much more complicated. In [J09], one of the authors introduced new ideas for unveiling the structure of highorder GJMS-operators. Among other things, these led to the discovery of a recursive structure among these operators [J13].

Special cases of recursive formulas for GJMS-operators are the relations

P_{4} − P_{2}^{2} = −4δ(Pd) + μ_{4}  (1.4)

for the Paneitz operator P_{4} and

P_{6} − (2P_{2}P_{4} + 2P_{4}P_{2} − 3P_{2}^{3}) = −48δ(P^{2}d) + μ_{6}  (1.5)

SUMMATION FORMULAS FOR GJMS-OPERATORS AND Q-CURVATURES 3

for the GJMS-operator P_{6} of locally conformally flat metrics. Here the scalar functions μ_{4} and μ_{6} are given in terms of the curvature of g; for explicit formulas see [J13].

In order to describe the general recursive formula for GJMS-operators, we need to introduce some notation. A sequence I = (I_{1}, …, I_{r}) of integers I_{j} ≥ 1 will be regarded as a composition of the sum |I| = I_{1} + I_{2} + ⋯ + I_{r}. As usual, a composition of a positive integer m is a representation of m as a sum m = m_{1} + m_{2} + ⋯ + m_{r} of positive integers m_{1}, m_{2}, …, m_{r}, where two representations which contain the same summands but differ in the order of the summands are regarded as different. |I| will be called the size of I. For any composition I, we set

P_{2I} = P_{2I_{1}} ∘ ⋯ ∘ P_{2I_{r}}.  (1.6)

For N ≥ 1, let

M_{2N} =^{def} ∑_{|I|=N} m_{I}P_{2I}  (1.7)

with the multiplicities m_{I} defined by

m_{I} = −(−1)^{r}|I|! (|I| − 1)! ∏_{j=1}^{r} \frac{1}{I_{j}! (I_{j} − 1)!} ∏_{j=1}^{r−1} \frac{1}{I_{j} + I_{j+1}} .  (1.8)

Here, an empty product has to be interpreted as 1. In particular, m_{(N)} = 1 for all N ≥ 1, and m_{I} = m_{I^{−1}}, where I^{−1} is the reverse composition of I = (I_{1}, I_{2}, …, I_{r}), that is, I^{−1} = (I_{r}, I_{r−1}, …, I_{1}).

Theorem 1.1 of [J13] implies that, for locally conformally flat Riemannian manifolds of dimension n ≥ 3, we have

M_{2N} = −c_{N}δ(P^{N−1}d) + zeroth-order part,  (1.9)

where

c_{N} = 2^{N−1}N! (N − 1)!.  (1.10)

Eq. (1.9) may be regarded as a summation formula for the operator M_{2N}. We note that the sum (1.7) contains 2^{N−1} terms. Since one of these terms is P_{2N}, the operator M_{2N} is of order 2N. But (1.9) shows that the operator M_{2N} is actually only secondorder. In other words, in the sum (1.7), huge cancellations take place. Since m_{(N)} = 1, Eq. (1.9) can be regarded as a formula for P_{2N}^{0} =^{def} P_{2N} − P_{2N}(1) in terms of a linear combination of the second-order operator δ(P^{N−1}d) and compositions of lower-order GJMS-operators. In that sense, it is a recursive formula for P_{2N}^{0}. Eqs. (1.4) and (1.5) are the first two special cases.

The proof of (1.9) in [J13] rests on the theory of residue families as developed in [J09]. Residue families are generalizations of certain differential intertwining operators in representation theory ([J09], [KOSS13]). It is the recursive structure of these families which is responsible for the identities (1.9). An alternative proof of (1.9) was given in [FG12b]. It rests on the original definition of the GJMS-operators in terms of the Fefferman–Graham ambient metric as given in [GJMS92]. Note that the formulations in [J10] and [J13] restrict to Riemannian metrics while in [FG12b] the signature of the metric is arbitrary.

Branson [B95] used representation theory to derive explicit expressions for the GJMS-operators of the spaces 𝕊^{q,p}. In the present paper, we use these formulas to give a direct proof of the relation (1.9) for 𝕊^{q,p}. More precisely, we prove the following summation formula.

Theorem 1.1. On 𝕊^{q,p}, we have

M_{4N} = (2N)! (2N −1)! (\frac{1}{2} −B^{2}−C^{2}) , N ≥ 1  (1.11)

and

M_{4N+2} = (2N +1)! (2N)! (−B^{2}+C^{2}), N ≥ 0.  (1.12)

Here

B^{2} = −Δ_{𝕊^{q}} + (\frac{q − 1}{2})^{2} and C^{2} = −Δ_{𝕊^{p}} + (\frac{p − 1}{2})^{2} .  (1.13)

Theorem 1.1 implies

M_{4N}^{0} = (2N)! (2N −1)! (Δ_{𝕊^{q}} + Δ_{𝕊^{p}}), N ≥ 1,
M_{4N+2}^{0} = (2N +1)! (2N)! (Δ_{𝕊^{q}} − Δ_{𝕊^{p}}), N ≥ 0

for the non-constant parts of the operators M_{2N}. But using

P = \frac{1}{2} (\begin{matrix}1_{𝕊^{q}} & 0 \\ 0 & −1_{𝕊^{p}} \end{matrix}) ,  (1.14)

these two identities can be written in the form

M_{4N}^{0} = −c_{2N}δ(P^{2N−1}d),
M_{4N+2}^{0} = −c_{2N+1}δ(P^{2N}d),

where c_{2N} and c_{2N+1} are defined in (1.10). Thus, Theorem 1.1 actually implies (1.9). Note that Theorem 1.1 also determines the zeroth-order parts of all M_{2N}. It is easy to verify that the result fits with the description of that part as given in [J13, Ex. 8.2].

We continue with the discussion of Q-curvatures. The main result of [J10] concerns the zeroth-order parts of the GJMS-operators. These define Branson’s Q-curvatures as follows. For even n, general metrics and 2N < n, the equation

P_{2N}(1) = (−1)^{N} (\frac{n}{2} − N) Q_{2N}

defines a scalar curvature quantity Q_{2N}. The quantities Q_{2}, …, Q_{n−2} are called the subcritical Q-curvatures. The critical Q-curvature Q_{n} can be defined through the subcritical ones by continuation in dimension. In odd dimensions as well as for locally conformally flat metrics in even dimensions, the Q-curvatures Q_{2N} are well defined for all N ≥ 1. For details we refer to [B95] and [J09].

Explicit formulas for Q_{2} and Q_{4} follow from (1.2) and (1.3):

Q_{2} = J and Q_{4} = \frac{n}{2}J^{2} − 2|P|^{2} − ΔJ.

For N ≥ 3, explicit formulas for Q_{2N} are substantially more complicated. Therefore, it is of some interest to establish recursive formulas for Q-curvatures. Such a formula was found in [J10].

In order to motivate its formulation, we start with the description of two special cases. First of all, we rewrite Q_{4} in the form

Q_{4} = −P_{2}(Q_{2}) − Q_{2}^{2} + 2! 2^{3} v_{4},  (1.15)

where

v_{4} = \frac{1}{4} tr(∧^{2}P).

Next, for locally conformally flat metrics, we have (see [J09])

Q_{6} = [−2P_{2}(Q_{4}) + 2P_{4}(Q_{2}) − 3P_{2}^{2}(Q_{2})] − 6 [Q_{4} + P_{2}(Q_{2})] Q_{2} − 2! 3! 2^{5} v_{6}, (1.16)

where

v_{6} = −\frac{1}{8} tr(∧^{3}P).

More conceptually, the coefficients v_{4} and v_{6} are Taylor coefficients of the function

v(r) = det (1 − \frac{r^{2}}{4} P) = 1 + v_{2}r^{2} + v_{4}r^{4} + v_{6}r^{6} + ⋯,  (1.17)

which describes the volume form of a Poincaré–Einstein metric associated to the locally conformally flat metric g (see [G00], [FG12a]). Eq. (1.17) implies

v_{2k} = (−\frac{1}{2})^{k} tr(∧^{k}P).

Now, one can prove that the formulas (1.15) and (1.16) are equivalent to the respective identities

Q_{4} + P_{2}(Q_{2}) = 2! 2^{4} w_{4}  (1.18)

and

−Q_{6} − 2P_{2}(Q_{4}) + 2P_{4}(Q_{2}) − 3P_{2}^{2}(Q_{2}) = 2! 3! 2^{6} w_{6},  (1.19)

where the quantities w_{4} and w_{6} are Taylor coefficients of

w(r) = \sqrt{v(r)},

i.e.,

(1 + r^{2}w_{2} + r^{4}w_{4} + r^{6}w_{6} + ⋯)^{2} = v(r).

More explicitly, we have

2w_{2} = v_{2},
2w_{4} = \frac{1}{4}(4v_{4} − v_{2}^{2}),
2w_{6} = \frac{1}{8}(8v_{6} − 4v_{4}v_{2} + v_{2}^{3}).

Now, in these terms, the main result of [J10] implies that

∑_{|I|=N} m_{I} \frac{P_{2I}(1)}{\frac{n}{2} − I_{last}} = N! (N −1)! w_{2N}  (1.20)

for locally conformally flat metrics and N ≥ 1. Here, I_{last} denotes the last entry of the composition I. For I = (J, \frac{n}{2}), the quotient in (1.20) is to be interpreted as

(−1)^{\frac{n}{2}} P_{2J}(Q_{n}).

Again, the sum in (1.20) contains 2^{N−1} terms. By separating the contribution for I = (N), Eq. (1.20) can be regarded as a formula for Q_{2N}. It expresses Q_{2N} in terms of lower-order Q-curvatures, lower-order GJMS-operators and w_{2N}. Eqs. (1.18) and (1.19) are the first two special cases.

The proof of (1.20) in [J10] rests on the recursive structure of residue families. This result played an important role in the discussion of the recursive structure of GJMS-operators in [J13]. Again, an alternative proof of (1.20) was given in [FG12b].

The following result proves (1.20) for 𝕊^{q,p} by proving the following summation formula.

Theorem 1.2. On 𝕊^{q,p}, we have

∑_{|I|=N} m_{I} \frac{P_{2I}(1)}{\frac{n}{2} − I_{last}} = N! (N −1)! ∑_{M=0}^{N}(−1)^{M} \binom{\frac{q}{2}}{M}\binom{\frac{p}{2}}{N −M}  (1.21)

for all N ≥ 1.

In order to see that Theorem 1.2 implies (1.20), we note that (1.14) and (1.17) imply

v(r) = (1 − r^{2}/4)^{q}(1 + r^{2}/4)^{p}.

Hence

w(r) = (1 − r^{2}/4)^{q/2}(1 + r^{2}/4)^{p/2}.

It follows that the right-hand side of (1.21) coincides with

N! (N − 1)! 2^{2N} w_{2N}.

The remaining part of the paper is organized as follows. In Section 2, we prove Theorem 1.1. Section 3 contains the proof of a summation formula which contains Theorem 1.2 as a special case. The proofs of some technical results which are used in the course of these proofs are collected in an appendix.

2. Proof of Theorem 1.1

We start with a description of the GJMS-operators on 𝕊^{q,p}.

Theorem 2.1 ([B95], Theorem 6.2). On 𝕊^{q,p}, the GJMS-operators are given by the product formulas

P_{4N} = ∏_{j=1}^{N}(B+C+(2j−1))(B−C−(2j−1))(B+C−(2j−1))(B−C+(2j−1)) (2.1)

and

P_{4N+2} = (−B^{2}+C^{2}) ∏_{j=1}^{N}(B+C+2j)(B−C−2j)(B+C−2j)(B−C+2j). (2.2)

Here, the operators B and C are defined as the positive square roots of the nonnegative operators B^{2} and C^{2}.

The reader should note that the formulas (2.1) and (2.2) are equivalent to the product representations

P_{4N} = ∏_{j=1}^{N} ((B^{2}−C^{2})^{2} − 2(2j−1)^{2}(B^{2}+C^{2}) + (2j−1)^{4})  (2.3)

and

P_{4N+2} = (−B^{2}+C^{2}) ∏_{j=1}^{N} ((B^{2}−C^{2})^{2} − 2(2j)^{2}(B^{2}+C^{2}) + (2j)^{4}) .  (2.4)

An advantage of the latter formulas is that they do not require to leave the framework of differential operators.

The formulas (2.1) and (2.2) can be restated uniformly as

P_{2N} = 2^{2N} ((C+B+1−N)/2)_{N} ((C−B+1−N)/2)_{N} ,  (2.5)

where (α)_{m} is the usual Pochhammer symbol defined by (α)_{m} = α(α+1) ⋯ (α+m−1) for m ≥ 1, and (α)_{0} = 1. In terms of the variables

X = C + B and Y = C − B,  (2.6)

the product formula (2.5) is equivalent to

P_{2N} = 2^{2N} ((X + 1 − N)/2)_{N} ((Y + 1 − N)/2)_{N} .  (2.7)

In the sequel, we shall regard P_{2N} as this polynomial in the variables X and Y.

In the proofs of Theorems 1.1 and 1.2, the basis (((X + 1 − A)/2)_{A})_{A=0,1,...} of the linear space of all polynomials in X will play an essential role. (A glance at (2.7) may suggest why this could be the case.) In the following lemma, we compute the structure coefficients with respect to multiplication of this basis of polynomials. They will enter the inductive proofs of Lemma 2.2 and of Theorems 2.2 and 3.1.

Lemma 2.1. For all non-negative integers A and B, we have

((X + 1 − A)/2)_{A} ((X + 1 − B)/2)_{B}

= ∑_{j=0}^{⌊(A+B)/2⌋}(−1)^{j} \frac{(−A/2)_{j} (−B/2)_{j} (−(A + B)/2)_{j}}{j!} ((X +1−A−B+2j)/2)_{A+B−2j} . (2.8)

Remark 2.1. The sum on the right-hand side of (2.8) terminates at the latest at j = ⌊(A + B)/2⌋. If A or B should be even, it terminates already at j = A/2, respectively at j = B/2.

Proof. The reader should recall the standard hypergeometric notation

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

where (a_{1})_{m}, etc. are again Pochhammer symbols. In terms of this notation, the right-hand side of (2.8) reads

((X + 1 − A − B)/2)_{A+B} _{3}F_{2}[\begin{matrix}−(A + B)/2, −A/2, −B/2 \\ (1 + X − A − B)/2, (1 − X − A − B)/2 \end{matrix}; 1] .

The _{3}F_{2}-series can be evaluated by means of the Pfaff–Saalschütz summation (cf. [S66, (2.3.1.3); Appendix (III.2)])

3F_{2}[\begin{matrix}a, b, −n \\ c, 1 + a + b − c − n \end{matrix}; 1] = \frac{(c − a)_{n} (c − b)_{n}}{(c)_{n} (c − a − b)_{n}} ,

where n is a non-negative integer. If we apply the formula, then we obtain the lefthand side of (2.8) after little simplification. □

Lemma 2.2. For all positive integers a < N, the partial sum

S(N, a) = ∑_{J:|J|+a=N} m_{(J,a)}P_{2J}  (2.9)

satisfies

S(N, a) = \binom{N − 1}{a − 1} ∑_{k=0}^{⌊(N−a)/2⌋} ∑_{l=0}^{⌊(N−a)/2⌋}(−1)^{N+k+l+a}2^{2N−2k−2l−2a}

· ((X + 1 − N + a + 2k)/2)_{N−a−2k} ((Y + 1 − N + a + 2l)/2)_{N−a−2l}

· \frac{(−N + a)_{2k} (−N + a)_{2l} (−N/2)_{k} (−N/2)_{l}}{k! l!} _{4}F_{3}[\begin{matrix}−\frac{1}{2}, −k, −l, \frac{1}{2} − \frac{N}{2} \\ −\frac{N}{2} , \frac{a}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.10)

Proof. We prove the claim by induction on N, the case N = 2 being straightforward to check. Suppose that we have already established (2.10) up to N −1. If a+|J| = N, we write m_{(a,J)} as

m_{(a,J)} = \begin{cases}−\frac{N! (N − 1)!}{(N − a)! (N − a − 1)!} \frac{1}{N · a! (a − 1)!}m_{(N−a)}, & if J = (N − a), \\ −\frac{N! (N − 1)!}{(N − a)! (N − a − 1)!} \frac{1}{(a + b) a! (a − 1)!}m_{(b,K)}, & if J = (b, K), \end{cases}

and, if J = (b, K), we write P_{2J} = P_{2b}P_{2K}. Then the left-hand side of (2.10) becomes

− \frac{N! (N − 1)!}{N · a! (a − 1)! (N − a)! (N − a − 1)!}P_{2(N−a)}

− ∑_{b=1}^{N−a−1} ∑_{K:b+|K|=N−a} \frac{N! (N − 1)!}{(a + b) a! (a − 1)! (N − a)! (N − a − 1)!}m_{(b,K)}P_{2b}P_{2K}. (2.11)

If we now use the induction hypothesis, then this sum simplifies to

− ∑_{b=1}^{N−a} \frac{N! (N − 1)!}{(a + b) a! (a − 1)! (N − a)! (N − a − 1)!}2^{2b}((X + 1 − b)/2)_{b} ((Y + 1 − b)/2)_{b}

· \binom{N − a − 1}{b − 1} ∑_{k=0}^{⌊(N−a−b)/2⌋} ∑_{l=0}^{⌊(N−a−b)/2⌋}(−1)^{N+k+l+a+b}2^{2N−2k−2l−2a−2b}

· ((X + 1 − N + a + b + 2k)/2)_{N−a−b−2k} ((Y + 1 − N + a + b + 2l)/2)_{N−a−b−2l}

· \frac{(−N + a + b)_{2k} (−N + a + b)_{2l} ((−N + a)/2)_{k} ((−N + a)/2)_{l}}{k! l!}

·_{4}F_{3}[\begin{matrix}−\frac{1}{2}, −k, −l, \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \\ \frac{a}{2} − \frac{N}{2} , \frac{a}{2} + \frac{b}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} + \frac{b}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.12)

Here, the term containing P_{2(N−a)}, appearing in (2.11), arises as the summand for b = N − a, since this forces k and l to be zero.

Now, Lemma 2.1 implies the expansions

((X + 1 − b)/2)_{b} ((X + 1 − N + a + b + 2k)/2)_{N−a−b−2k}

= ∑_{j_{1}=0}^{⌊(N−a−2k)/2⌋}(−1)^{j_{1}} \frac{(−b/2)_{j_{1}} (−(N − a − b − 2k)/2)_{j_{1}} (−(N − a − 2k)/2)_{j_{1}}}{j_{1}!}

· ((X + 1 − N + a + 2k + 2j_{1})/2)_{N+a−2k−2j_{1}}.

and

((Y + 1 − b)/2)_{b} ((Y + 1 − N + a + b + 2l)/2)_{N−a−b−2l}

= ∑_{j_{2}=0}^{⌊(N−a−2l)/2⌋}(−1)^{j_{2}} \frac{(−b/2)_{j_{2}} (−(N − a − b − 2l)/2)_{j_{2}} (−(N − a − 2l)/2)_{j_{2}}}{j_{2}!}

· ((Y + 1 − N + a + 2l + 2j_{2})/2)_{N+a−2l−2j_{2}}.

We use these in (2.12) and, in addition, perform the index transformation s_{1} = k+j_{1} and s_{2} = l + j_{2}. Thus, we arrive at the expression

− ∑_{s_{1}=0}^{⌊(N−a)/2⌋} ∑_{s_{2}=0}^{⌊(N−a)/2⌋} ((X +1−N +a+2s_{1})/2)_{N−a−2s_{1}} ((Y +1−N +a+2s_{2})/2)_{N−a−2s_{2}}

· ∑_{b=1}^{N−a} \frac{N! (N − 1)!}{(a + b) a! (a − 1)! (N − a)! (N − a − 1)!}\binom{N − a − 1}{b − 1}(−1)^{N+s_{1}+s_{2}+a+b}2^{2N−2a}

· ∑_{k=0}^{⌊(N−a−b)/2⌋} ∑_{l=0}^{⌊(N−a−b)/2⌋}(−b/2)_{s_{1}−k} (−b/2)_{s_{2}−l} ((−N + a)/2)_{s_{1}} ((−N + a)/2)_{s_{2}}

· ((−N + a + b)/2)_{s_{1}} ((−N + a + b)/2)_{s_{2}} \frac{((−N + a + b + 1)/2)_{k} ((−N + a + b + 1)/2)_{l}}{k! l! (s_{1} − k)! (s_{2} − l)!}

·_{4}F_{3}[\begin{matrix}−\frac{1}{2}, −k, −l, \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \\ \frac{a}{2} − \frac{N}{2} , \frac{a}{2} + \frac{b}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} + \frac{b}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.13)

In this expression, we now concentrate on the terms involving the summation index k only:

∑_{k=0}^{s_{1}} \frac{(−b/2)_{s_{1}−k} (−(N − a − b − 1)/2)_{k} (−k)_{s}}{k! (s_{1} − k)!} ,  (2.14)

where s stands for the summation index of the _{4}F_{3}-series in (2.19). Because of the term (−k)_{s} in the numerator of the summand, we may start the summation at k = s (instead of at k = 0). Hence, if we write this sum in hypergeometric notation, we obtain

\frac{(−b/2)_{s_{1}−s} (−(N − a − b − 1)/2)_{s} (−s)_{s}}{s!}_{2}F_{1}[\begin{matrix}−s_{1} + s, \frac{1}{2} + \frac{a}{2} − \frac{N}{2} + \frac{b}{2} + s \\ 1 + \frac{b}{2} − s_{1} + s \end{matrix}; 1] .

This _{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula (see [S66, (1.7.7), Appendix (III.4)]), so that the sum in (2.14) equals

\frac{(−s_{1})_{s} (−(N − a − b − 1)/2)_{s} ((1 + a − N)/2)_{s_{1}}}{s_{1}! ((1 + a − N)/2)_{s}} .

Clearly, an analogous computation can be done for the sum over l in (2.13). Altogether, we see that (2.13) simplifies to

− ∑_{s_{1}=0}^{⌊(N−a)/2⌋} ∑_{s_{2}=0}^{⌊(N−a)/2⌋} ((X +1−N +a+2s_{1})/2)_{N−a−2s_{1}} ((Y +1−N +a+2s_{2})/2)_{N−a−2s_{2}}

· ∑_{b=1}^{N−a}(−1)^{N+s_{1}+s_{2}+a+b}2^{2N−2s_{1}−2s_{2}−2a} \frac{N! (N − 1)! \binom{N−a−1}{b−1}}{(a + b) a! (a − 1)! (N − a)! (N − a − 1)!}

· \frac{(−N + a)_{2s_{1}} (−N + a)_{2s_{2}} ((−N + a + b)/2)_{s_{1}} ((−N + a + b)/2)_{s_{2}}}{s_{1}! s_{2}!}

·_{4}F_{3}[\begin{matrix}−\frac{1}{2}, −s_{1}, −s_{2}, \frac{1}{2} + \frac{a}{2} + \frac{b}{2} − \frac{N}{2} \\ \frac{a}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} − \frac{N}{2} , \frac{a}{2} + \frac{b}{2} − \frac{N}{2} \end{matrix} ; 1]

= − ∑_{s_{1}=0}^{⌊(N−a)/2⌋} ∑_{s_{2}=0}^{⌊(N−a)/2⌋} ((X+1−N +a+2s_{1})/2)_{N−a−2s_{1}} ((Y +1−N +a+2s_{2})/2)_{N−a−2s_{2}}

· ∑_{b=1}^{N−a}(−1)^{N+s_{1}+s_{2}+a+b}2^{2N−2s_{1}−2s_{2}−2a} \frac{N! (N − 1)!}{(a + b) a! (a − 1)! (N − a)! (b − 1)!}

· ∑_{s=0}^{s_{1}} \frac{(−N + a)_{2s_{1}} (−N + a)_{2s_{2}} ((−N + a + b + 2s)/2)_{s_{1}−s} ((−N + a + b + 2s)/2)_{s_{2}−s}}{s_{1}! s_{2}!}

· \frac{(−1/2)_{s} (−s_{1})_{s} (−s_{2})_{s} (N − a − 2s)!}{s! (N − a)! (N − a − b − 2s)!}. (2.15)

Next we concentrate on the terms involving the summation index b only:

\frac{1}{(N − a − 2s − 1)!} ∑_{b=1}^{N−a} \frac{(−1)^{b}}{a + b} \binom{N − a − 2s − 1}{b − 1}

· ((−N + a + b + 2s)/2)_{s_{1}−s} ((−N + a + b + 2s)/2)_{s_{2}−s}.

By Lemma A.2 with X = −a − 1 and M = N − a − 2s − 1, this is equal to

\frac{(−1)^{N−a−2s−1}}{(−N + 2s)_{N−a−2s}} ((−N + 2s)/2)_{s_{1}−s}((−N + 2s)/2)_{s_{2}−s}

+ χ(s_{1} = s_{2} = (N − a)/2) · 2^{−N+a+2s}

= −\frac{a!}{(N − 2s)!} \frac{(−N/2)_{s_{1}} (−N/2)_{s_{2}}}{(−N/2)_{s}^{2}} + χ(s_{1} = s_{2} = (N − a)/2) · 2^{−N+a+2s},

where χ(S) = 1 if S is true and χ(S) = 0 otherwise. If we substitute this in (2.15), then we obtain

∑_{s_{1}=0}^{⌊(N−a)/2⌋} ∑_{s_{2}=0}^{⌊(N−a)/2⌋} ((X +1−N +a+2s_{1})/2)_{N−a−2s_{1}} ((Y +1−N +a+2s_{2})/2)_{N−a−2s_{2}}

· (−1)^{N+s_{1}+s_{2}+a}2^{2N−2s_{1}−2s_{2}−2a} \frac{(N − 1)!}{(a − 1)! (N − a)!}

· \frac{(−N + a)_{2s_{1}} (−N + a)_{2s_{2}} (−N/2)_{s_{1}} (−N/2)_{s_{2}}}{s_{1}! s_{2}!}

· ∑_{s=0}^{s_{1}} \frac{(−1/2)_{s} (−s_{1})_{s} (−s_{2})_{s} N! (N − a − 2s)!}{s! (N − 2s)! (−N/2)_{s}^{2} (N − a)!}

+ χ(N − a is even) · \frac{N! (N − 1)!}{a! (a − 1)! ((N − a)/2)!^{2}}

× ∑_{s=0}^{(N−a)/2} \frac{2^{−N+a+2s}(−1/2)_{s} (−(N − a)/2)_{s}^{2} (N − a − 2s)!}{s!}.

Here, the first sum is, upon rewriting, exactly equal to the right-hand side of (2.10) (except that s_{1} and s_{2} took over the role of k and l). On the other hand, if we write the second sum in hypergeometric notation, we obtain

χ(N − a is even) · \frac{N! (N − 1)!}{a! (a − 1)! ((N − a)/2)!^{2}}

× ∑_{s=0}^{(N−a)/2} \frac{2^{−N+a+2s}(−1/2)_{s} (−(N − a)/2)_{s}^{2} (N − a − 2s)!}{s!}

= χ(N − a is even) · \frac{N! (N − 1)!}{a! (a − 1)! ((N − a)/2)!^{2}} 2^{−N+a} (N − a)!_{2}F_{1}[\begin{matrix}−\frac{1}{2}, \frac{a}{2} − \frac{N}{2} \\ \frac{a}{2} − \frac{N}{2} + \frac{1}{2} \end{matrix}; 1] .

The _{2}F_{1}-series can be evaluated by the Chu–Vandermonde summation formula, so that the above expression becomes

χ(N − a is even) · \frac{N! (N − 1)!}{a! (a − 1)! ((N − a)/2)!^{2}} 2^{−N+a} (N − a)!\frac{((a − N + 2)/2)_{(N−a)/2}}{((a − N + 1)/2)_{(N−a)/2}} ,

which vanishes because of the term ((a − N + 2)/2)_{(N−a)/2}. This concludes the induction step. □

Theorem 2.2. For all positive integers N, we have

∑_{|I|=N} m_{I}P_{2I} = \begin{cases}N! (N − 1)! XY, & if N is odd, \\ \frac{1}{2}N! (N − 1)! (1 − X^{2} − Y^{2}) , & if N is even, \end{cases}  (2.16)

where P_{2N} is defined in (2.7).

Proof. By the definition of S(N, a) in (2.9), we have

∑_{|I|=N} m_{I}P_{2I} = P_{2N} + ∑_{a=1}^{N−1} S(N, a)P_{2a}.  (2.17)

If, in this equation, we substitute the expressions for P_{2a} and S(N, a) given by (2.7) and (2.10), then we see that the left-hand side of (2.16) is equal to

∑_{a=1}^{N} ((X + 1 − a)/2)_{a} ((Y + 1 − a)/2)_{a}\binom{N − 1}{a − 1}

· ∑_{k=0}^{⌊(N−a)/2⌋} ∑_{l=0}^{⌊(N−a)/2⌋}(−1)^{N+k+l+a}2^{2N−2k−2l}

· ((X + 1 − N + a + 2k)/2)_{N−a−2k} ((Y + 1 − N + a + 2l)/2)_{N−a−2l}

· \frac{(−N + a)_{2k} (−N + a)_{2l} (−N/2)_{k} (−N/2)_{l}}{k! l!} _{4}F_{3}[\begin{matrix}−\frac{1}{2}, −k, −l, \frac{1}{2} − \frac{N}{2} \\ −\frac{N}{2} , \frac{a}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.18)

Here, the term P_{2N}, appearing on the left-hand side of (2.16), arises as the summand for a = N, since this forces k and l to be zero.

Lemma 2.1 implies the expansions

((X + 1 − a)/2)_{a} ((X + 1 − N + a + 2k)/2)_{N−a−2k}

= ∑_{j_{1}=0}^{⌊(N−2k)/2⌋}(−1)^{j_{1}} \frac{(−a/2)_{j_{1}} (−(N − a − 2k)/2)_{j_{1}} (−(N − 2k)/2)_{j_{1}}}{j_{1}!}

· ((X + 1 − N + 2k + 2j_{1})/2)_{N−2k−2j_{1}}.

and ((Y + 1 − a)/2)_{a} ((Y + 1 − N + a + 2l)/2)_{N−a−2l}

= ∑_{j_{2}=0}^{⌊(N−2l)/2⌋}(−1)^{j_{2}} \frac{(−a/2)_{j_{2}} (−(N − a − 2l)/2)_{j_{2}} (−(N − 2l)/2)_{j_{2}}}{j_{2}!}

· ((Y + 1 − N + 2l + 2j_{2})/2)_{N−2l−2j_{2}}.

We use these in (2.18) and, in addition, perform the index transformation s_{1} = k + j_{1} and s_{2} = l + j_{2}. Thus, the left-hand side in (2.16) can be written in the form

∑_{s_{1}=0}^{⌊N/2⌋} ∑_{s_{2}=0}^{⌊N/2⌋} ∑_{a=1}^{N}(−1)^{N+s_{1}+s_{2}+a}2^{2N−2k−2l}\binom{N − 1}{a − 1}

· ((X + 1 − N + 2s_{1})/2)_{N−2s_{1}}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}}

· ∑_{k=0}^{⌊(N−a)/2⌋} ∑_{l=0}^{⌊(N−a)/2⌋} \frac{(−a/2)_{s_{1}−k} (−a/2)_{s_{2}−l} (−(N − a)/2)_{s_{1}} (−(N − a)/2)_{s_{2}}}{(s_{1} − k)! (s_{2} − l)! (−(N − a)/2)_{k} (−(N − a)/2)_{l}}

· \frac{(−N + a)_{2k} (−N + a)_{2l} (−N/2)_{s_{1}} (−N/2)_{s_{2}}}{k! l!} _{4}F_{3}[\begin{matrix}−\frac{1}{2}, −k, −l, \frac{1}{2} − \frac{N}{2} \\ −\frac{N}{2} , \frac{a}{2} − \frac{N}{2} , \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.19)

In this expression, we now concentrate on the terms involving the summation index k only:

∑_{k=0}^{⌊(N−a)/2⌋} 2^{−2k} \frac{(−a/2)_{s_{1}−k} (−N + a)_{2k} (−k)_{s}}{k! (s_{1} − k)! (−(N − a)/2)_{k}}

= ∑_{k=0}^{s_{1}} \frac{(−a/2)_{s_{1}−k} (−(N − a − 1)/2)_{k} (−k)_{s}}{k! (s_{1} − k)!} , (2.20)

where s stands for the summation index of the _{4}F_{3}-series in (2.19). Because of the term (−k)_{s} in the numerator of the summand, we may start the summation at k = s (instead of at k = 0). Hence, if we write this sum in hypergeometric notation, we obtain

\frac{(−a/2)_{s_{1}−s} (−(N − a − 1)/2)_{s} (−s)_{s}}{s!}_{2}F_{1}[\begin{matrix}−s_{1} + s, \frac{1}{2} − \frac{N}{2} + \frac{a}{2} + s \\ 1 + \frac{a}{2} − s_{1} + s \end{matrix}; 1] .

This _{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula, so that the sum in (2.20) equals

\frac{(−s_{1})_{s} (−(N − a − 1)/2)_{s} ((1 − N)/2)_{s_{1}}}{s_{1}! ((1 − N)/2)_{s}} .

Clearly, an analogous computation can be done for the sum over l in (2.19). Altogether, we see that (2.19) simplifies to

∑_{s_{1}=0}^{⌊N/2⌋} ∑_{s_{2}=0}^{⌊N/2⌋} ∑_{a=1}^{N}(−1)^{N+s_{1}+s_{2}+a}2^{2N} \binom{N − 1}{a − 1}

· (−N/2)_{s_{1}} ((1 − N)/2)_{s_{1}} (−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· ((X + 1 − N + 2s_{1})/2)_{N−2s_{1}}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}}

· \frac{(−(N − a)/2)_{s_{1}} (−(N − a)/2)_{s_{2}}}{s_{1}! s_{2}!}_{4}F_{3}[\begin{matrix}−\frac{1}{2}, −s_{1}, −s_{2}, \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \\ −\frac{N}{2} , \frac{1}{2} − \frac{N}{2} , \frac{a}{2} − \frac{N}{2} \end{matrix} ; 1] . (2.21)

Next we concentrate on the terms involving the summation index a only:

∑_{a=1}^{N}(−1)^{a}\binom{N − 1}{a − 1}\frac{(−(N − a)/2)_{s_{1}} (−(N − a)/2)_{s_{2}} (−(N − a − 1)/2)_{s}}{(−(N − a)/2)_{s}}

= \frac{(N − 1)!}{2^{2s} (N − 2s − 1)!} ∑_{a=1}^{N−2s}(−1)^{a}\binom{N − 2s − 1}{a − 1}

· (−(N − a − 2s)/2)_{s_{1}−s} (−(N − a − 2s)/2)_{s_{2}−s}

= −\frac{(N − 1)!}{2^{2s} (N − 2s − 1)!} ∑_{a=0}^{N−2s−1}(−1)^{a}\binom{N − 2s − 1}{a}

· ((a − N + 2s + 1)/2)_{s_{1}−s} ((a − N + 2s + 1)/2)_{s_{2}−s}, (2.22)

where, in abuse of notation, we wrote again s for the summation index of the_{4}F_{3}-series in (2.21). By Lemma A.1, this sum vanishes if the degree of

((a − N + 2s + 1)/2)_{s_{1}−s} ((a − N + 2s + 1)/2)_{s_{2}−s}

as a polynomial in a should be less than N − 2s − 1. In fact, this is almost always the case since s_{1} − s + s_{2} − s ≤ 2⌊N/2⌋ − 2s ≤ N − 2s. More precisely, the sum in (2.22) is non-zero only if:

(1) N is odd and s_{1} = s_{2} = (N − 1)/2;

(2) N is even and s_{1} = s_{2} = N/2 or {s_{1}, s_{2}} = {N/2, (N − 2)/2}. We now discuss these cases separately.

(1) Let N be odd. Then, as we observed above, in the sum (2.21) only the terms corresponding to s_{1} = s_{2} = (N − 1)/2 survive. Using Lemma A.1, we see that (2.21) reduces to

− 4 (−N)_{N−1}^{2} (X/2) (Y/2)

× ∑_{s=0}^{(N−1)/2}(−1)^{N−2s}2^{−N+2s+1} \frac{(N − 1)!}{2^{2s} ((N − 1)/2)!^{2}} \frac{(−1/2)_{s} ((1 − N)/2)_{s}}{s! (−N/2)_{s}}

= 2^{1−N}XY \frac{N!^{2} (N − 1)!}{((N − 1)/2)!^{2}}_{2}F_{1}[\begin{matrix}−\frac{1}{2}, \frac{1}{2} − \frac{N}{2} \\ −\frac{N}{2} \end{matrix} ; 1] .

The _{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula. After some simplification, we obtain N! (N − 1)! XY, in accordance with the claim in (2.16).

(2) Let N be even. As we observed above, there are now three terms in (2.21) which survive: the terms for s_{1} = s_{2} = N/2, for (s_{1}, s_{2}) = (N/2, (N − 2)/2), and for (s_{1}, s_{2}) = ((N − 2)/2, N/2).

We begin with the term for s_{1} = N/2 and s_{2} = (N − 2)/2. By Lemma A.1, this term equals

− 4 (−N)_{N} (−N)_{N−2} ((Y − 1)/2)_{2}

× ∑_{s=0}^{(N−2)/2}(−1)^{N−2s}2^{−N+2s+1} \frac{(N − 1)!}{2^{2s} (N/2)! ((N − 2)/2)!} \frac{(−1/2)_{s} ((2 − N)/2)_{s}}{s! ((1 − N)/2)_{s}}

= −2^{−N}(Y^{2} − 1)\frac{N!^{2} (N − 1)!}{(N/2)! ((N − 2)/2)!}_{2}F_{1}[\begin{matrix}−\frac{1}{2}, 1 − \frac{N}{2} \\ \frac{1}{2} − \frac{N}{2} \end{matrix} ; 1] .

Again, the_{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula. After some simplification, we obtain −\frac{1}{2}N! (N − 1)! (Y^{2} − 1). Clearly, an analogous computation yields that the term for s_{1} = (N − 2)/2 and s_{2} = N/2 equals −\frac{1}{2}N! (N − 1)! (X^{2} − 1). Finally, we consider the term in (2.21) corresponding to s_{1} = s_{2} = N/2. Here, in order to apply Lemma A.1, we need to compute the coefficient c_{N−2s−1} in the expansion

((a − N + 2s + 1)/2)_{(N−2s)/2} ((a − N + 2s + 1)/2)_{(N−2s)/2} = ∑_{k=0}^{N−2s} c_{k}\binom{a}{k}. (2.23)

We have c_{N−2s} = 2^{−N+2s}(N − 2s)!. Comparison of coefficients of a^{N−2s−1} on both sides of (2.23) then yields that

c_{N−2s−1} = 2^{−N+2s} (−2 ∑_{ℓ=1}^{(N−2s)/2}(2ℓ − 1) + ∑_{ℓ=1}^{N−2s−1} ℓ)

= 2^{−N+2s} (−2 (\frac{N − 2s}{2})^{2} + \frac{(N − 2s)(N − 2s − 1)}{2})

= −2^{−N+2s−1}(N − 2s).

Hence, if we use this, together with Lemma A.1, we obtain that the term corresponding to s_{1} = s_{2} = N/2 in (2.21) equals

(−N)_{N}^{2} ∑_{s=0}^{N/2}(−1)^{N−2s+1}2^{−N+2s−1}(N − 2s)\frac{(N − 1)!}{2^{2s} (N/2)!^{2}} \frac{(−1/2)_{s} (−N/2)_{s}}{s! ((1 − N)/2)_{s}}

= −2^{−N−1} \frac{N!^{2} (N − 1)!}{(N/2)!^{2}} (N _{2}F_{1}[\begin{matrix}−\frac{1}{2}, −\frac{N}{2} \\ \frac{1}{2} − \frac{N}{2} \end{matrix} ; 1] + \frac{N}{N − 1} _{2}F_{1}[\begin{matrix}\frac{1}{2}, 1 − \frac{N}{2} \\ \frac{3}{2} − \frac{N}{2} \end{matrix} ; 1]) .

Both _{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula. As it turns out, the first one sums to zero. If the result is simplified, one obtains −\frac{1}{2}N! (N − 1)! eventually.

Putting our results together, we obtain

\frac{1}{2}N! (N − 1)!(−X^{2} + 1 − Y^{2} + 1 − 1) = \frac{1}{2}N! (N − 1)!(1 − X^{2} − Y^{2})

for the left-hand side of (2.16), which is in accordance with the right-hand side. □

Now, Theorem 1.1 follows from Theorem 2.2 by recalling (2.6).

3. Proof of Theorem 1.2

As in Section 2 we regard all GJMS-operators as polynomials in the variables X and Y (see (2.7)). In particular, P_{2N} is divisible by the monomial (X + 1 − N).

Theorem 3.1. For all positive integers N, we have

∑_{a=1}^{N} ∑_{|J|=N−a} m_{(J,a)} \frac{1}{X + 1 − a}P_{2J}P_{2a}

= N!(N − 1)! ∑_{k=0}^{N}(−1)^{k}\binom{(X − Y + 1)/2}{k}\binom{(X + Y + 1)/2}{N − k}. (3.1)

Proof. By the definition of S(N, a) in (2.9), we have

∑_{a=1}^{N} ∑_{|J|=N−a} m_{(J,a)} \frac{1}{X + 1 − a}P_{2J}P_{2a} = \frac{1}{X + 1 − N} P_{2N} + ∑_{a=1}^{N−1} \frac{1}{X + 1 − a}S(N, a)P_{2a}.

The reader should note that the only difference to (2.17) are the factors 1/(X +1−N) in front of P_{2N} and the factor 1/(X + 1 − a) in front of the summand of the sum over a. We may therefore proceed as in the proof of Theorem 2.2, as long as the summation over a does not come into play. More precisely, by comparing with (2.21), we see that the left-hand side of (3.1) simplifies to

∑_{s_{1}=0}^{⌊N/2⌋} ∑_{s_{2}=0}^{⌊N/2⌋} ∑_{a=1}^{N}(−1)^{N+s_{1}+s_{2}+a}2^{2N} \binom{N − 1}{a − 1}

· (−N/2)_{s_{1}} ((1 − N)/2)_{s_{1}} (−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· \frac{1}{X + 1 − a}((X + 1 − N + 2s_{1})/2)_{N−2s_{1}}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}}

· \frac{(−(N − a)/2)_{s_{1}} (−(N − a)/2)_{s_{2}}}{s_{1}! s_{2}!}_{4}F_{3}[\begin{matrix}−\frac{1}{2}, −s_{1}, −s_{2}, \frac{1}{2} + \frac{a}{2} − \frac{N}{2} \\ −\frac{N}{2} , \frac{1}{2} − \frac{N}{2} , \frac{a}{2} − \frac{N}{2} \end{matrix} ; 1] . (3.2)

As in the proof of Theorem 2.2, we now concentrate on the terms involving the summation index a only:

∑_{a=1}^{N}(−1)^{a} \frac{1}{X + 1 − a}\binom{N − 1}{a − 1}\frac{(−(N − a)/2)_{s_{1}} (−(N − a)/2)_{s_{2}} (−(N − a − 1)/2)_{s}}{(−(N − a)/2)_{s}}

= \frac{(N − 1)!}{2^{2s} (N − 2s − 1)!} ∑_{a=1}^{N−2s}(−1)^{a} \frac{1}{X + 1 − a}\binom{N − 2s − 1}{a − 1}

· (−(N − a − 2s)/2)_{s_{1}−s} (−(N − a − 2s)/2)_{s_{2}−s}

= −\frac{(N − 1)!}{2^{2s} (N − 2s − 1)!} ∑_{a=0}^{N−2s−1}(−1)^{a} \frac{1}{X − a}\binom{N − 2s − 1}{a}

· ((a − N + 2s + 1)/2)_{s_{1}−s} ((a − N + 2s + 1)/2)_{s_{2}−s},

where, in abuse of notation, we wrote again s for the summation index of the_{4}F_{3}-series in (3.2). By Lemma A.2 with M = N − 2s − 1, this is equal to

\frac{(−1)^{N−2s} (N − 1)!}{2^{2s}(X − N + 2s + 1)_{N−2s}} ((X + 1 − N + 2s)/2)_{s_{1}−s}((X + 1 − N + 2s)/2)_{s_{2}−s}

− χ(s_{1} = s_{2} = N/2) · 2^{−N}(N − 1)!. (3.3)

We substitute this in (3.2). As before in the proof of Lemma 2.2, the second term in (3.3) does not contribute anything since the (remaining) sum over s vanishes. Therefore, substitution of (3.3) in (3.2) leads to the expression

∑_{s_{1}=0}^{⌊N/2⌋} ∑_{s_{2}=0}^{⌊N/2⌋} ∑_{s=0}^{s_{1}}(−1)^{s_{1}+s_{2}+s}2^{2N−2s}(−N/2)_{s_{1}} ((1 − N)/2)_{s_{1}} (−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· \frac{((X + 1 − N + 2s)/2)_{N−s−s_{1}}((X + 1 − N + 2s)/2)_{s_{2}−s}}{(s_{1} − s)! (X − N + 2s + 1)_{N−2s}}

· ((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}} \frac{(N − 1)! (−1/2)_{s} (−s_{2})_{s}}{s_{2}! s! (−N/2)_{s} ((1 − N)/2)_{s}} .

We write the sum over s_{1} in hypergeometric notation to obtain

∑_{s_{2}=0}^{⌊N/2⌋} ∑_{s=0}^{s_{1}}(−1)^{s_{2}}2^{2N−2s}(−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· \frac{((X + 1 − N + 2s)/2)_{N−2s}((X + 1 − N + 2s)/2)_{s_{2}−s}}{(X − N + 2s + 1)_{N−2s}}

· ((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}} \frac{(N − 1)! (−1/2)_{s} (−s_{2})_{s}}{s_{2}! s!}_{2}F_{1}[\begin{matrix}−\frac{N}{2} + s, \frac{1}{2} − \frac{N}{2} + s \\ −\frac{X}{2} + \frac{1}{2} − \frac{N}{2} + s \end{matrix}; 1] .

The _{2}F_{1}-series can be evaluated by means of the Chu–Vandermonde summation formula. After some simplification, we arrive at

∑_{s_{2}=0}^{⌊N/2⌋} ∑_{s=0}^{s_{1}}(−1)^{s_{2}}2^{N}(−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· ((X + 1 − N + 2s)/2)_{s_{2}−s}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}} \frac{(N − 1)! (−1/2)_{s} (−s_{2})_{s}}{s_{2}! s!}

= ∑_{s_{2}=0}^{⌊N/2⌋}(−1)^{s_{2}}2^{N} \frac{(N − 1)!}{s_{2}!}(−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· ((X + 1 − N)/2)_{s_{2}}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}} _{2}F_{1}[\begin{matrix}−\frac{1}{2}, −s_{2} \\ \frac{X}{2} + \frac{1}{2} − \frac{N}{2} \end{matrix} ; 1] .

After having evaluated the _{2}F_{1}-series by means of the Chu–Vandermonde summation formula, we are left with

∑_{s_{2}=0}^{⌊N/2⌋}(−1)^{s_{2}}2^{N} \frac{(N − 1)!}{s_{2}!}(−N/2)_{s_{2}} ((1 − N)/2)_{s_{2}}

· ((X + 2 − N)/2)_{s_{2}}((Y + 1 − N + 2s_{2})/2)_{N−2s_{2}}

= 2^{N}(N − 1)!((Y + 1 − N)/2)_{N} _{3}F_{2}[\begin{matrix}\frac{X}{2} − \frac{N}{2} + 1, −\frac{N}{2} , \frac{1}{2} − \frac{N}{2} \\ −\frac{Y}{2} − \frac{N}{2} + \frac{1}{2}, \frac{Y}{2} − \frac{N}{2} + \frac{1}{2} \end{matrix}; 1] .

By the transformation formula in Lemma A.3, this is equal to

(N − 1)! ((X + Y − 2N + 3)/2)_{N} _{2}F_{1}[\begin{matrix}−\frac{X}{2} + \frac{Y}{2} − \frac{1}{2}, −N \\ \frac{X}{2} + \frac{Y}{2} − N + \frac{3}{2} \end{matrix}; −1] .

which agrees with the right-hand side of (3.1). This finishes the proof. □

Now we are finally in the position to establish Theorem 1.2. Proof of Theorem 1.2. Using n = q + p, we find

X(1) = (C + B)(1) = \frac{n}{2} − 1 and Y (1) = (C − B)(1) = \frac{p − q}{2} .

Thus, we can write the constant term of P_{2N} in the form

P_{2N} (\frac{n}{2} − 1, \frac{p − q}{2} ) .

Now we calculate

∑_{|I|=N} m_{I} \frac{1}{\frac{n}{2} − I_{last}} P_{2I}(1)

= ∑_{a=1}^{N} ∑_{|J|=N−a} m_{(J,a)} \frac{1}{\frac{n}{2} − a}P_{2J} (\frac{n}{2} − 1, \frac{p − q}{2} ) P_{2a} (\frac{n}{2} − 1, \frac{p − q}{2} )

= N!(N − 1)! ∑_{k=0}^{N}(−1)^{k}\binom{\frac{q}{2}}{k}\binom{\frac{p}{2}}{N − k}.

In the last step we have applied Theorem 3.1. □

SUMMATION FORMULAS FOR GJMS-OPERATORS AND Q-CURVATURES 19

Appendix. Proofs of some auxiliary results

Here we prove three technical lemmas which were used in Sections 2 and 3. Lemma A.1. Let p(x) be a polynomial in x, and suppose that

p(x) = ∑_{k≥0} c_{k}\binom{x}{k},

for (uniquely determined) coefficients c_{k}. Furthermore, let M be an integer. Then

∑_{a=0}^{M}(−1)^{a}\binom{M}{a}p(a) = (−1)^{M}c_{M}.

Proof. This is a classical fact from finite difference calculus. For the convenience of the reader, we give the simple proof. We calculate

∑_{a=0}^{M}(−1)^{a}\binom{M}{a}p(a) = ∑_{a=0}^{M}(−1)^{a}\binom{M}{a} ∑_{k≥0} c_{k}\binom{a}{k}

= ∑_{k≥0} c_{k}\binom{M}{k} ∑_{a=0}^{M}(−1)^{a}\binom{M − k}{a − k}

= ∑_{k≥0} c_{k}\binom{M}{k}(−1)^{k}δ_{M,k} = (−1)^{M}c_{M}.

The proof is complete. □ Lemma A.2. Let p(x) be a polynomial in x, and suppose that

p(x) = ∑_{k≥0} c_{k}\binom{x}{k}

for (uniquely determined) c_{k}. Furthermore, let M be an integer. If the degree of p(x) is at most M, then

∑_{b=0}^{M} \frac{(−1)^{b}}{X − b}\binom{M}{b}p(b) = (−1)^{M} \frac{M!}{(X − M)_{M+1}} p(X).

If the degree of p(x) equals M + 1, then

∑_{b=0}^{M} \frac{(−1)^{b}}{X − b}\binom{M}{b}p(b) = (−1)^{M} \frac{M!}{(X − M)_{M+1}} p(X) + \frac{c_{M+1}}{M + 1}.

Proof. It suffices to verify the claim for a basis of the vector space of polynomials in x of degree at most M + 1. We choose \{\binom{x}{k}\}_{0≤k≤M+1} as such a basis. If k ≤ M, we have

∑_{b=0}^{M} \frac{(−1)^{b}}{X − b}\binom{M}{b}\binom{b}{k} = \frac{(−1)^{k}}{X − k} \binom{M}{k}_{2}F_{1}[\begin{matrix}−M + k, k − X \\ k − X + 1 \end{matrix}; 1]

= \frac{(−1)^{k}}{X − k} \binom{M}{k}\frac{(M − k)!}{(−X + k + 1)_{M−k}}

= (−1)^{M} \frac{M!}{k!} \frac{(X − k + 1)_{k}}{(X − M)_{M+1}}

= (−1)^{M} \frac{M!}{(X − M)_{M+1}} \binom{X}{k},

while if k = M + 1 we obtain 0. Here, we used the Chu–Vandermonde summation formula to evaluate the _{2}F_{1}-series. All this is in agreement with our claims. □

Lemma A.3. For all non-negative integers N, we have

3F_{2}[\begin{matrix}a, \frac{1}{2} − \frac{N}{2} , −\frac{N}{2} \\ e, 1 − N − e \end{matrix}; 1] = 2^{−N} \frac{(e − a)_{N}}{(e)_{N}} _{2}F_{1}[\begin{matrix}1 − a − e − N, −N \\ 1 + a − e − N \end{matrix}; −1] .  (A.1)

Proof. In [KR03, (3.14)],

4F_{3}[\begin{matrix}a, b, \frac{1}{2} − \frac{N}{2} , −\frac{N}{2} \\ \frac{1}{2} + a + b, e, 1 − N − e \end{matrix}; 1] = \frac{(a + e)_{N}}{(e)_{N}} _{3}F_{2}[\begin{matrix}2 a, a + b, −N \\ 2 a + 2 b, a + e \end{matrix}; 1] ,

we let b tend to infinity. As a result, we obtain the transformation formula

3F_{2}[\begin{matrix}a, \frac{1}{2} − \frac{N}{2} , −\frac{N}{2} \\ e, 1 − N − e \end{matrix}; 1] = \frac{(a + e)_{N}}{(e)_{N}} _{2}F_{1}[\begin{matrix}2 a, −N \\ a + e \end{matrix}; \frac{1}{2}] .

If we now apply the transformation formula (see [S66, (1.7.1.3)])

2F_{1}[\begin{matrix}A, B \\ C \end{matrix}; z] = (1 − z)^{−B}_{2}F_{1}[\begin{matrix}C − A, B \\ C \end{matrix}; −\frac{z}{1 − z} ]

to the _{2}F_{1}-series on the right-hand side, then we obtain

2^{−N} \frac{(a + e)_{N}}{(e)_{N}} _{2}F_{1}[\begin{matrix}e − a, −N \\ a + e \end{matrix}; −1] .

Finally, if we reverse the order of summands in this _{2}F_{1}-series (that is, if we denote the summation index in the _{2}F_{1}-series by s, then we replace s by N − s and rewrite the result again in hypergeometric notation), then we arrive at the right-hand side of (A.1). □

References

[B95] T. Branson, Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc. 347 (1995), 3671–3742.

[CKOP11] J.-L. Clerc, T. Kobayashi, B. Ørsted and M. Pevzner, Generalized Bernstein–Reznikov integrals. Math. Ann. 349, no. 2 (2011), 395–431. arXiv:0906.2874.

[FG12a] C. Fefferman and C. R. Graham, The Ambient Metric. Annals of Math. Studies 178, Princeton University Press, 2012. arXiv:0710.0919v2

[FG12b] C. Feffermann and C. R. Graham, Juhl’s formulae for GJMS-operators and Q-curvatures. J. Amer. Math. Soc. 26, no. 4 (2013), 1191–1207. arXiv:1203.0360.

[GH04] A. R. Gover and K. Hirachi. Conformally invariant powers of the Laplacian – a complete nonexistence theorem. J. Amer. Math. Soc. 17, no. 2 (2004), 389–405. arXiv:math/

         0304082v2.

[GJMS92] C. R. Graham, R. Jenne, L. J. Mason and G. A. J. Sparling. Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2) 46 no. 3 (1992), 557–565.

[G92] C. R. Graham. Conformally invariant powers of the Laplacian. II. Nonexistence. J. London Math. Soc. (2) 46 no. 3 (1992), 566–576.

[G00] C. R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo (2) Suppl. 63, (2000), 31–42. arXiv:math/9909042v1.

[J09] A. Juhl. Families of Conformally Covariant Differential Operators, Q-Curvature and Holography, volume 275 of Progress in Mathematics. Birkhäuser–Verlag, 2009.

[J10] A. Juhl, On the recursive structure of Branson’s Q-curvature. arXiv:1004.1784v2.

[J13] A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures. Geom. Funct. Anal. 23 (2013), 1278–1370. arXiv:1108.0273.

[K13] T. Kobayashi, F-method for symmetry breaking operators, Differential Geom. Appl. 33, Supplement, 272–289 (2014). arXiv:1303.3541.

[KOSS13] T. Kobayashi, B. Orsted, P. Somberg and V. Soucek, Branching laws for Verma modules and applications in parabolic geometry. I. arXiv:1305.6040.

[KR03] C. Krattenthaler and K. Srinivasa Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math. 160 (2003), 159–173.

[S66] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.

Humboldt-Universität, Institut für Mathematik, Unter den Linden, D-10099 Berlin

  E-mail address: ajuhl@math.hu-berlin.de

Universität Wien, Fakultät für Mathematik, Nordbergstrasze 15, A-1090 Wien

  http://www.mat.univie.ac.at/~kratt/