MODULARITY AND TOTAL ELLIPTICITY OF SOME MULTIPLE SERIES OF HYPERGEOMETRIC TYPE

V.P. SPIRIDONOV

Abstract. We collect some new evidence on the validity of the conjecture that every totally elliptic hypergeometric series is modular invariant.

The following general definition of elliptic hypergeometric functions was proposed in [20].

Definition 1. The formal (multiple) series

∑_{λ_{1},... ,λ_{n}=0}^{∞} c(λ_{1}, …, λ_{n}) or ∑_{0≤λ_{1}≤...≤λ_{n}≤∞} c(λ_{1}, …, λ_{n})  (1)

define elliptic hypergeometric functions if:

1) the coefficients c(λ) are symmetric with respect to an action of the permutation group S_{n} upon the summation variables λ_{1}, …, λ_{n} and the free parameters entering c(λ);

2) for all ℓ = 1, …, n the ratios

h_{ℓ}(λ) = \frac{c(λ_{1}, …, λ_{ℓ} + 1, …, λ_{n})}{c(λ_{1}, …, λ_{ℓ}, …, λ_{n})},  (2)

are elliptic functions of λ_{ℓ}, ℓ = 1, …, n, considered as continuous complex variables.

It is convenient to take the initial condition c(0, …, 0) = 1, so that the coefficients c(λ_{1}, …, λ_{n}), λ_{j} ∈ ℕ, are uniquely determined from the functions h_{ℓ}(λ) by solving n first order recurrence relations fixed in (2).

For n = 1, we start from a general elliptic function of order r + 1, which can be factorized as follows [24]:

h(λ) = z \frac{θ_{1}(u_{0} + λ, …, u_{r} + λ; σ, τ)}{θ_{1}(v_{0} + λ, …, v_{r} + λ; σ, τ)} = z \frac{θ(t_{0}q^{λ}, …, t_{r}q^{λ}; p)}{θ(w_{0}q^{λ}, …, w_{r}q^{λ}; p)},  (3)

where

θ_{1}(u_{0}, …, u_{k}; σ, τ) = ∏_{i=0}^{k} θ_{1}(u_{i}; σ, τ),  θ(t_{0}, …, t_{k}; p) = ∏_{i=0}^{k} θ(t_{i}; p), (4)

with p = e^{2πiτ} , Im(τ) > 0, q = e^{2πiσ}, and

θ_{1}(u; σ, τ) = −i ∑_{n=−∞}^{∞}(−1)^{n}p^{(2n+1)^{2}/8}q^{(n+1/2)u}
= p^{1/8}iq^{−u/2} (p; p)_{∞} θ(q^{u}; p), u ∈ ℂ,

Date: July 14, 2002.

being one of the standard Jacobi theta functions. The function θ(z; p) has the infinite product representation

θ(z; p) = (z; p)_{∞}(pz^{−1}; p)_{∞},  (a; p)_{∞} = ∏_{n=0}^{∞}(1 − ap^{n}),  (5)

and obeys the following properties

θ(pz; p) = θ(z^{−1}; p) = −z^{−1}θ(z; p).  (6)

Sometimes it is convenient to drop for brevity bases q, p or modular parameters σ, τ in the notations for theta functions or elliptic shifted factorials. We shall use also the convenient elliptic numbers notation [3]:

[u; σ, τ] ≡ θ_{1}(u; σ, τ), or [u] ≡ θ_{1}(u).

The parameters u_{i}, v_{i}, t_{i}, w_{i} in (3) are connected to each other as follows

t_{i} = q^{u_{i}}, w_{i} = q^{v_{i}}

and satisfy the constraints

∑_{i=0}^{r}(u_{i} − v_{i}) = 0, or ∏_{i=0}^{r} t_{i} = ∏_{i=0}^{r} w_{i},  (7)

which guarantee that the meromorphic function h(λ) is double periodic:

h(λ + σ^{−1}) = h(λ), h(λ + τσ^{−1}) = h(λ).

Remind, that the function [u] is entire, odd [−u] = −[u], and doubly quasiperiodic

[u + σ^{−1}] = −[u],  [u + τσ^{−1}] = −e^{−πiτ−2πiσu}[u].  (8)

The modular P SL(2, ℤ)-group,

τ → \frac{aτ + b}{cτ + d},  σ → \frac{σ}{cτ + d},  (9)

where a, b, c, d ∈ ℤ and ad−bc = 1, is generated by two transformations: τ → τ +1, σ → σ, and τ → −τ^{−1}, σ → στ^{−1}. In these two cases we have

[u; σ, τ + 1] = e^{πi/4}[u; σ, τ],  (10)
[u; σ/τ, −1/τ] = i(−iτ)^{1/2}e^{πiσ^{2}u^{2}/τ} [u; σ, τ],  (11)

and the sign of (−iτ)^{1/2} is fixed from the positivity of its real part condition.

Place one of the poles of h(λ) in (3) into the point λ = −1 by setting v_{0} = 1 or w_{0} = q (this is just a convenient normalization). The resulting function h(λ) generates the single variable elliptic hypergeometric series, which we present in the multiplicative system of notations:

r+1E_{r} ({t_{0}, …, t_{r} \atop w_{1}, …, w_{r}}; q, p; z) = ∑_{λ=0}^{∞} \frac{θ(t_{0}, t_{1}, …, t_{r}; p; q)_{λ}}{θ(q, w_{1}, …, w_{r}; p; q)_{λ}} z^{λ}  (12)

with the elliptic shifted factorials defined as follows

θ(t_{0}, …, t_{k}; p; q)_{λ} ≡ ∏_{m=0}^{k} ∏_{j=0}^{λ−1} θ(t_{m}q^{j}; p).

In the additive form, elliptic shifted factorials take the form

[u_{0}, …, u_{k}]_{λ} ≡ ∏_{m=0}^{k} ∏_{j=0}^{λ−1}[u_{m} + j; σ, τ].

Series (12) are well defined even without ellipticity constraint (7). Therefore, it is convenient to refer to the general hypergeometric type series built out from the Jacobi theta functions as theta hypergeometric series. In this case, the equalities (7) are called the balancing condition.

It is possible to extend definition (1)-(2) by relaxing the double periodicity condition to a quasiperiodicity, which is characteristic to the Jacobi theta functions (8). As a result, we get theta functions analogues of the plain and basic generalized hypergeometric series _{s}F_{r} and _{s}Φ_{r} without constraints upon the parameters like (7). For further details, we refer to [20].

Theta hypergeometric series in (12) are called totally elliptic if the series coefficients c(λ) are elliptic not only in the variable λ, but in all free parameters among u_{i}, v_{i} as well. As shown in [20], the requirement of total ellipticity leads to the following additional constraints upon the parameters:

u_{0} + 1 = u_{1} + v_{1} = … = u_{r} + v_{r}, or qt_{0} = t_{1}w_{1} = … = t_{r}w_{r},  (13)

which are well-known as the well-poisedness conditions for plain and basic hypergeometric series. This gives a kind of explanation of the origin of this notion itself.

At the elliptic level, the very-well-poisedness condition differs from the ones used in the plain and basic hypergeometric cases [11]. Namely, it demands that, in addition to the constraints (13), four parameters are placed into four points of the second order of the elliptic function h(λ) around a distinguished point:

u_{r−3} = \frac{1}{2}u_{0} + 1, u_{r−2} = \frac{1}{2}u_{0} + 1 − \frac{1}{2σ} ,

u_{r−1} = \frac{1}{2}u_{0} + 1 − \frac{τ}{2σ} , u_{r} = \frac{1}{2}u_{0} + 1 + \frac{1 + τ}{2σ} ,  (14)

or, in the multiplicative form,

t_{r−3} = t_{0}^{1/2}q, t_{r−2} = −t_{0}^{1/2}q,
t_{r−1} = t_{0}^{1/2}qp^{−1/2}, t_{r} = −t_{0}^{1/2}qp^{1/2}.  (15)

As a result, the very-well-poised theta hypergeometric series _{r+1}E_{r} take the form

r+1E_{r} = ∑_{λ=0}^{∞} \frac{θ(t_{0}q^{2λ}; p)}{θ(t_{0}; p)} ∏_{m=0}^{r−4} \frac{θ(t_{m}; p; q)_{λ}}{θ(qt_{0}/t_{m}; p; q)_{λ}} (−qz)^{λ}.  (16)

Series of such type (with a different counting of the number of parameters and z = −1) have been considered for the first time by Frenkel and Turaev [9]. Their consideration was inspired by a rich set of exactly solvable statistical mechanics models built by Date et al [3]. In a different context, within a program on systematic reconsideration of special functions of one variable as “self-similar” functions (see, e.g. the review [19]), elliptic hypergeometric series have been derived by Zhedanov and the author as solutions of particular three term recurrence relation describing new families of biorthogoanl rational functions [22]. For p → 0, series (16) reduce to the very-well-poised _{r−1}Φ_{r−2} basic hypergeometric series [11] of the argument −qz (note the shift of the index r by −2).

Frenkel and Turaev [9] proved the following summation formula:

∑_{λ=0}^{N} q^{λ} \frac{θ(t_{0}q^{2λ}; p)}{θ(t_{0}; p)} ∏_{r=0}^{5} \frac{θ(t_{r}; p; q)_{λ}}{θ(qt_{0}t_{r}^{−1}; p; q)_{λ}}

= \frac{θ(qt_{0}; p; q)_{N} ∏_{1≤r<s≤3} θ(qt_{0}/t_{r}t_{s}; p; q)_{N}}{θ(qt_{0}/t_{1}t_{2}t_{3}; p; q)_{N} ∏_{r=1}^{3} θ(qt_{0}/t_{r}; p; q)_{N}} ,  (17)

where ∏_{i=1}^{5} t_{i} = qt_{0}^{2} and t_{4} = q^{−N}, N ∈ ℕ. In the classification introduced in [20], this corresponds to a closed form expression for the terminating very-well-poised balanced _{10}E_{9} theta hypergeometric series at z = −1.

In the multivariable case, the general form of the elliptic hypergeometric series is not established yet. There are several types of such series in the literature leading to explicit summation formulae, generalizing (17). They extend the rich set of identities for series of hypergeometric type that were found first by Biedenharn, Holman, and Louck (see, e.g. [10]) and lifted further to the basic hypergeometric series level by Milne (see, e.g. [13, 14]) and Gustafson (see, e.g. [4, 12]). The first two multiple elliptic hypergeometric series were proposed by Warnaar [23], for one of which he has proved an extension of the Frenkel-Turaev sum. For the second type of series, Warnaar conjectured a similar summation formula. It was proved at the plain and q-hypergeometric series levels by van Diejen and the author in [6] (some simplified versions of this sum were proven earlier in [5]). A partial proof of the elliptic sum was given as well. It was proved completely by Rosengren with the help of a different technique in [15].

The third type of series was proposed in [7] together with an elliptic extension of the multiple Jackson sum of [4, 14], which was proved partially on the basis of modular group invariance property. In the present paper, we consider two more types of multiple series, analyze some of their properties, discuss the corresponding Jackson type summation formulae, and show their partial validity using the modularity arguments. In an independent detailed study, Rosengren [16] arrived to the latter sums as well, proved them (together with the sum proposed in [7]), and analyzed some of their consequences.

As in the single summation series example, multiple theta hypergeometric series are called totally elliptic if their coefficients c(λ) are elliptic in the summation indices λ_{ℓ} together with all free parameters entering the arguments of θ_{1}-functions. The main purpose of the present paper consists in providing new evidence of the validity of the following conjecture.

Conjecture 2. Every totally elliptic hypergeometric series is automatically modular invariant.

For n = 1 series (in the general situation) and C_{n} sums discussed in [6, 7, 15, 23], this statement was proved in [20]. We prove it below for the very first example of multiple elliptic hypergeometric series summation formula found by Warnaar in [23] (it was missed inadvertently in the analysis of [20]) and for two above-mentioned new examples of elliptic hypergeometric series. This note should be considered as a complement to [20].

In [23], Warnaar proved the following elliptic analogue of the Schlosser [18] multivariable C_{n} Jackson sum:

∑_{λ_{1},... ,λ_{n}=0}^{N} q^{∑_{j=1}^{n} jλ_{j}} ∏_{1≤i<j≤n} \frac{θ(t_{i}t_{j}^{−1}q^{λ_{i}−λ_{j}}, t_{i}t_{j}q^{λ_{i}+λ_{j}}; p)}{θ(t_{i}t_{j}^{−1}, t_{i}t_{j}; p)}

× ∏_{j=1}^{n} \frac{θ(t_{j}^{2}q^{2λ_{j}}; p)θ(t_{j}^{2}, bt_{j}, ct_{j}, dt_{j}, et_{j}, q^{−N}; p; q)_{λ_{j}}}{θ(t_{j}^{2}; p)θ(q, t_{j}q/b, t_{j}q/c, t_{j}q/d, t_{j}q/e, t_{j}^{2}q^{N+1}; p; q)_{λ_{j}}}  (18)

= ∏_{1≤i<j≤n} \frac{θ(t_{i}t_{j}q^{N}; p)}{θ(t_{i}t_{j}; p)} ∏_{j=1}^{n} \frac{θ(t_{j}^{2}q, q^{2−j}/bc, q^{2−j}/bd, q^{2−j}/cd; p; q)_{N}}{θ(t_{j}^{−1}q^{2−n}/bcd, t_{j}q/b, t_{j}q/c, t_{j}q/d; p; q)_{N}} ,

where parameters satisfy the balancing constraint bcde = q^{N−n+2}.

Remark 1. The summation formulae given in [18, 23] contain one more parameter a, but it can be removed by rescaling the variables t_{j}, b, c, d, e.

Theorem 3. Theta hypergeometric series appearing at the left-hand side of (18) is totally elliptic and modular invariant.

Proof. Consider ratios of successive series coefficients in (18):

h_{ℓ}(λ) = q^{ℓ} \frac{θ(t_{ℓ}^{2}q^{2λ_{ℓ}+2}; p)}{θ(t_{ℓ}^{2}q^{2λ_{ℓ}}; p)} ∏_{i=1}^{ℓ−1} \frac{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}−1}, t_{i}t_{ℓ}q^{λ_{i}+λ_{ℓ}+1}; p)}{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}}, t_{i}t_{ℓ}q^{λ_{i}+λ_{ℓ}}; p)}

× ∏_{j=ℓ+1}^{n} \frac{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}+1}, t_{ℓ}t_{j}q^{λ_{ℓ}+λ_{j}+1}; p)}{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}}, t_{ℓ}t_{j}q^{λ_{ℓ}+λ_{j}}; p)}  (19)

×\frac{θ(t_{ℓ}^{2}q^{λ_{ℓ}}, bt_{ℓ}q^{λ_{ℓ}}, ct_{ℓ}q^{λ_{ℓ}}, dt_{ℓ}q^{λ_{ℓ}}, et_{ℓ}q^{λ_{ℓ}}, q^{−N+λ_{ℓ}}; p)}{θ(q^{λ_{ℓ}+1}, t_{ℓ}q^{λ_{ℓ}+1}/b, t_{ℓ}q^{λ_{ℓ}+1}/c, t_{ℓ}q^{λ_{ℓ}+1}/d, t_{ℓ}q^{λ_{ℓ}+1}/e, t_{ℓ}^{2}q^{N+1+λ_{ℓ}}; p)}.

Prove first that this function is elliptic in λ_{k}. For this, it is sufficient to replace q^{λ_{k}} by pq^{λ_{k}}, apply the rule θ(pz; p) = −z^{−1}θ(z; p), and to check cancellation of all extra factors appearing from that. For k < ℓ or k > ℓ, the situation is very simple and we skip its consideration. For k = ℓ, the first theta functions fraction in (19) yields the factor q^{−4}, the next two products, q^{2(1−n)}. The last fraction involving parameters b, …, e generates the product q^{2N+6}(bcde)^{−2}. As a result, the whole resulting multiplier is equal to 1 because of the balancing condition.

To consider ellipticity in the parameters and modular invariance of h_{ℓ}(λ), make the replacements t_{j}, b, c, d, e → q^{g_{j}}, q^{β}, q^{γ}, q^{δ}, q^{ϵ}. Then we can rewrite (19) in terms of the elliptic numbers as follows

h_{ℓ}(λ) = \frac{[2g_{ℓ} + 2λ_{ℓ} + 2]}{[2g_{ℓ} + 2λ_{ℓ}]} ∏_{i=1,i≠ℓ}^{n} \frac{[g_{i} − g_{ℓ} + λ_{i} − λ_{ℓ} − 1, g_{i} + g_{ℓ} + λ_{i} + λ_{ℓ} + 1]}{[g_{i} − g_{ℓ} + λ_{i} − λ_{ℓ}, g_{i} + g_{ℓ} + λ_{i} + λ_{ℓ}]}

×\frac{[2g_{ℓ} + λ_{ℓ}, β + g_{ℓ} + λ_{ℓ}, γ + g_{ℓ} + λ_{ℓ}]}{[λ_{ℓ} + 1, g_{ℓ} + λ_{ℓ} + 1 − β, g_{ℓ} + λ_{ℓ} + 1 − γ]}

×\frac{[δ + g_{ℓ} + λ_{ℓ}, ϵ + g_{ℓ} + λ_{ℓ}, λ_{ℓ} − N]}{[g_{ℓ} + λ_{ℓ} + 1 − δ, g_{ℓ} + λ_{ℓ} + 1 − ϵ, 2g_{ℓ} + N + 1 + λ_{ℓ}]}.  (20)

From this representation it follows that, if we rewrite (18) in terms of the elliptic numbers, then the first factor q^{∑_{j=1}^{n} jλ_{j}} disappears. As a result, simultaneous permutation of the variables λ_{j}, λ_{k} and parameters g_{j}, g_{k} does not change the series coefficients c(λ) (there are equal numbers of sign changes in the numerator and denominator of c(λ) stemming from the antisymmetry [−u] = −[u]). Thus, a symmetric group invariance is naturally implemented inside these series.

Ellipticity in g_{k} of (20) follows from the invariance of (19) under the shift t_{k} → pt_{k}. For k < ℓ or k > ℓ this is easy to verify. For k = ℓ, the first fraction of theta functions yields the multiplier q^{−4}, the next two products give q^{2(1−n)}. The rest of h_{ℓ} generates the multiplier q^{2N+6}(bcde)^{−2}, so that the total multiplication factor becomes equal to 1 due to the balancing condition. In order to check ellipticity in β (or, equivalently, in γ or δ), make replacements b → pb, e → p^{−1}e in (19). The only contributions come from the last line expression and, since θ(p^{−1}z; p) = −p^{−1}zθ(z; p), they are seen to cancel each other.

Turn now to modular invariance. Since there is an equal number of θ_{1}-functions in the numerator and denominator of h_{ℓ}, the numerical factors appearing from the transformations τ → τ + 1, τ → −1/τ in (10), (11) cancel out. Thus, it is sufficient to verify that the difference of squares of arguments of elliptic numbers in the numerator and denominator vanishes. For the first fraction of theta functions in (20), such a difference is equal to 4 + 8(g_{ℓ} + λ_{ℓ}). The next product yields 2(1 + 2(g_{ℓ} + λ_{ℓ}))(n − 1). The rest of h_{ℓ} generates the difference

2(β + γ + δ + ϵ)(1 + 2(g_{ℓ} + λ_{ℓ})) − 4 − 4(g_{ℓ} + λ_{ℓ})(N + 3) − 2(N + 1).

The total sum of these expressions is equal to zero, which proves modular invariance of h_{ℓ}-functions. In the Eichler-Zagier terminology [8], this means that all functions h_{ℓ}(λ), ℓ = 1, …, n, in (20) are meromorphic Jacobi forms of the weight 0 and index 0. All together they form a structure associated with the root system C_{n}.

Although coincidence of the left and right hand sides of (18) was proved in [23], for completeness, we have verified that the expression on the right-hand side of (18) is an elliptic function of all parameters g_{j}, β, γ, δ and it is modular invariant.

During the work on the elliptic Bailey chain [21], the author came across the Bhatnagar-Schlosser paper [2], where various Bailey transformation formulae for multiple _{10}Φ_{9} series were discussed. The Bailey chain built in [21] deals with an infinite sequence of such transformations for single variable elliptic hypergeometric series. A comparison of [2, 21] has naturally lead to a conjecture that much of the corresponding results of [2] should have elliptic analogues. In particular, this conjecture assumed the validity of the following elliptic generalizations of the Milne [13], Schlosser [17], and Bhatnagar [1] multivariable analogues of the Jackson sum.

An elliptic Milne A_{n} sum:

∑_{{0≤λ_{j} ≤N_{j} \atop j=1,... ,N_{j}}} q^{∑_{j=1}^{n} jλ_{j}} ∏_{j=1}^{n} \frac{θ(t_{j}q^{λ_{j}+|λ|}; p)}{θ(t_{j}; p)} ∏_{1≤i<j≤n} \frac{θ(t_{i}t_{j}^{−1}q^{λ_{i}−λ_{j}}; p)}{θ(t_{i}t_{j}^{−1}; p)}

× ∏_{i,j=1}^{n} \frac{θ(t_{i}t_{j}^{−1}q^{−N_{j}}; p; q)_{λ_{i}}}{θ(qt_{i}t_{j}^{−1}; p; q)_{λ_{i}}} ∏_{j=1}^{n} \frac{θ(t_{j}; p; q)_{|λ|}}{θ(t_{j}q^{1+N_{j}}; p; q)_{|λ|}}

×\frac{θ(b, c; p; q)_{|λ|}}{θ(q/d, q/e; p; q)_{|λ|}} ∏_{j=1}^{n} \frac{θ(dt_{j}, et_{j}; p; q)_{λ_{j}}}{θ(t_{j}q/b, t_{j}q/c; p; q)_{λ_{j}}}

= \frac{θ(q/bd, q/cd; p; q)_{|N|}}{θ(q/d, q/bcd; p; q)_{|N|}} ∏_{j=1}^{n} \frac{θ(t_{j}q, t_{j}q/bc; p; q)_{N_{j}}}{θ(t_{j}q/b, t_{j}q/c; p; q)_{N_{j}}} ,  (21)

where |λ| = λ_{1} + ⋯ + λ_{n}, |N| = N_{1} + ⋯ + N_{n}, and bcde = q^{1+|N|}.

An elliptic Bhatnagar-Schlosser D_{n} sum (as stated in [2], the D_{n} sums discussed in [1, 17] are essentially equivalent to each other):

∑_{{0≤λ_{j} ≤N_{j} \atop j=1,... ,N_{j}}} q^{∑_{j=1}^{n} jλ_{j}} ∏_{j=1}^{n} \frac{θ(t_{j}q^{λ_{j}+|λ|}; p)}{θ(t_{j}; p)} ∏_{1≤i<j≤n} \frac{θ(t_{i}t_{j}^{−1}q^{λ_{i}−λ_{j}}; p)}{θ(t_{i}t_{j}^{−1}; p)}

× ∏_{1≤i<j≤n} θ(t_{i}t_{j}q/bcd; p; q)_{λ_{i}+λ_{j}}^{−1} ∏_{i,j=1}^{n} θ(t_{i}t_{j}q^{1+N_{j}}/bcd; p; q)_{λ_{i}}

× ∏_{i,j=1}^{n} \frac{θ(t_{i}t_{j}^{−1}q^{−N_{j}}; p; q)_{λ_{i}}}{θ(qt_{i}t_{j}^{−1}; p; q)_{λ_{i}}} ∏_{j=1}^{n} \frac{θ(t_{j}; p; q)_{|λ|}θ(bcdt_{j}^{−1}; p; q)_{|λ|−λ_{j}}}{θ(t_{j}q^{1+N_{j}}, bcdt_{j}^{−1}q^{−N_{j}}; p; q)_{|λ|}}

×\frac{θ(b, c, d; p; q)_{|λ|}}{∏_{j=1}^{n} θ(t_{j}q/b, t_{j}q/c, t_{j}q/d; p; q)_{λ_{j}}}

= ∏_{j=1}^{n} \frac{θ(t_{j}q, t_{j}q/bc, t_{j}q/bd, t_{j}q/cd; p; q)_{N_{j}}}{θ(t_{j}q/bcd, t_{j}q/b, t_{j}q/c, t_{j}q/d; p; q)_{N_{j}}} .  (22)

Remark 2. In the presentation of Milne A_{n} sum given in the appendix of [2] there is an extra parameter a, but it can be removed by rescaling the variables t_{j}, d, and e. Similarly, in the Schlosser version of the D_{n} sum given there, the extra parameter a can be removed by rescaling t_{j} (the same is true for the Bhatnagar version of this sum).

Theorem 4. The A_{n} and D_{n} theta hypergeometric series standing on the left-hand sides of the equalities (21) and (22) are totally elliptic and modular invariant. The corresponding right-hand side expressions obey the same properties.

Proof. Consider the functions h_{ℓ}(λ) in the A_{n} case:

h_{ℓ}(λ) = q^{ℓ} ∏_{{j=1 \atop j≠ℓ}}^{n} \frac{θ(t_{j}q^{λ_{j}+|λ|+1}; p)}{θ(t_{j}q^{λ_{j}+|λ|}; p)} \frac{θ(t_{ℓ}q^{λ_{ℓ}+|λ|+2}; p)}{θ(t_{ℓ}q^{λ_{ℓ}+|λ|}; p)}

× ∏_{i=1}^{ℓ−1} \frac{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}−1}; p)}{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}}; p)} ∏_{j=ℓ+1}^{n} \frac{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}+1}; p)}{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}}; p)}

× ∏_{j=1}^{n} \frac{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−N_{j}}, t_{j}q^{|λ|}; p)}{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}+1}, t_{j}q^{1+N_{j}+|λ|}; p)}

×\frac{θ(bq^{|λ|}, cq^{|λ|}, dt_{ℓ}q^{λ_{ℓ}}, et_{ℓ}q^{λ_{ℓ}}; p)}{θ(q^{|λ|+1}/d, q^{|λ|+1}/e, t_{ℓ}q^{λ_{ℓ}+1}/b, t_{ℓ}q^{λ_{ℓ}+1}/c; p)}.  (23)

Prove first ellipticity of h_{ℓ}(λ) in λ_{k}. For k < ℓ or k > ℓ, the shift q^{λ_{k}} → pq^{λ_{k}} in h_{ℓ} gives the following extra multipliers: q^{−n} and q^{−2} from the first two fractions of theta functions, respectively; the next two products yield q; from the product ∏_{j=1}^{n} on the third line we get the contribution q^{n+|N|}; and the rest of h_{ℓ} generates the factor q^{1−|N|}. The total multiplier is equal to 1.

Similarly, replacing q^{λ_{ℓ}} by pq^{λ_{ℓ}}, we get successively the multipliers q^{1−n}, q^{−4}, q^{1−ℓ}, q^{ℓ−n}, q^{2n+2|N|}, and q^{2−2|N|}, whose total product equals 1. Thus the coefficients c(λ) in the left-hand side of (21) are elliptic functions of all λ_{k}.

In order to see that there exists a natural action of the symmetric group, we make substitutions t_{j}, b, c, d, e → q^{g_{j}}, q^{β}, q^{γ}, q^{δ}, q^{ϵ}, and rewrite sum (21) in terms of the elliptic numbers:

∑_{{0≤λ_{j} ≤N_{j} \atop j=1,... ,N_{j}}} ∏_{j=1}^{n} \frac{[g_{j} + λ_{j} + |λ|]}{[g_{j}]} ∏_{1≤i<j≤n} \frac{[g_{i} − g_{j} + λ_{i} − λ_{j}]}{[g_{i} − g_{j}]}

× ∏_{i,j=1}^{n} \frac{[g_{i} − g_{j} − N_{j}]_{λ_{i}}}{[1 + g_{i} − g_{j}]_{λ_{i}}} ∏_{j=1}^{n} \frac{[g_{j}]_{|λ|}}{[g_{j} + 1 + N_{j}]_{|λ|}}

×\frac{[β, γ]_{|λ|}}{[1 − δ, 1 − ϵ]_{|λ|}} ∏_{j=1}^{n} \frac{[δ + g_{j}, ϵ + g_{j}]_{λ_{j}}}{[g_{j} + 1 − β, g_{j} + 1 − γ]_{λ_{j}}}  (24)

= \frac{[1 − β − δ, 1 − γ − δ]_{|N|}}{[1 − δ, 1 − β − γ − δ]_{|N|}} ∏_{j=1}^{n} \frac{[1 + g_{j}, 1 + g_{j} − β − γ]_{N_{j}}}{[g_{j} + 1 − β, g_{j} + 1 − γ]_{N_{j}}} .

Consider the permutation of (λ_{k}, g_{k}, N_{k}) with (λ_{m}, g_{m}, N_{m}) for some fixed k and m. There are only two factors which require some consideration. The first one, given by the product ∏_{1≤i<j≤n} appears to be symmetric due to an equal number of sign changes in the numerator and denominator (in the same way as it was for the sum (18)). The product ∏_{i,j=1}^{n} appears to be symmetric because it is taken over all possible values of both indices i and j. Thus, according the definition given at the beginning, we are dealing with particular elliptic hypergeometric series.

Prove now the total ellipticity of functions h_{ℓ}. Ellipticity in β (or γ), or the invariance of (23) under the shifts b → pb, e → p^{−1}e, follows from the cancellation of multipliers −b^{−1}q^{−|λ|}, −p^{−1}et_{ℓ}q^{λ_{ℓ}} and −q^{|λ|+1}e^{−1}, −pbt_{ℓ}^{−1}q^{−λ_{ℓ}−1} coming from the numerator and denominator of the last line expression, respectively. Similar situation holds for the shifts d → pd, e → p^{−1}e, which proves ellipticity in δ.

An essentially more complicated computation is needed in order to see, that c(λ) in (24) is modular invariant. Consider for this the difference of squares of arguments of elliptic numbers in the numerator and denominator of h_{ℓ}. The first ratio of theta functions gives n−1+2(|g|−g_{ℓ} +n|λ|−λ_{ℓ}), where |g| = g_{1} +…+g_{n}. The second ratio yields 4(1 + |λ| + λ_{ℓ} + g_{ℓ}). The next two products generate the difference n − 1 + 2n(g_{ℓ} + λ_{ℓ}) − 2|g| − 2|λ|. The product ∏_{j=1}^{n} of ratios of theta functions contributes −2(n + |N|)(1 + g_{ℓ} + λ_{ℓ} + |λ|). For the rest of h_{ℓ}, we get

(β + |λ|)^{2} + (γ + |λ|)^{2} + (δ + g_{ℓ} + λ_{ℓ})^{2} + (ϵ + g_{ℓ} + λ_{ℓ})^{2} − (|λ| + 1 − δ)^{2}

−(|λ| + 1 − ϵ)^{2} − (g_{ℓ} + λ_{ℓ} + 1 − β)^{2} − (g_{ℓ} + λ_{ℓ} + 1 − γ)^{2}.

The total sum of all these cumbersome expressions is equal to zero, which means that h_{ℓ} is invariant under transformations (10), (11) generating the full modular group.

As far as the total ellipticity and modularity of the expression standing on the right-hand side of (24) are concerned, they are easily checked. Indeed, the invariance of the right-hand side of (21) under the shifts t_{j} → pt_{j} or d → pd (i.e., ellipticity in g_{j} or δ) is obvious. The shift b → pb generates from the elliptic shifted factorials θ(…)_{|N|} the factor c^{|N|}, which is cancelled by the extra multipliers coming from the elliptic shifted factorials θ(…)_{N_{j}}. By symmetry, this means ellipticity in both β and γ.

As to the difference of squares of the arguments of elliptic numbers in the numerator and denominator of the right-hand side of (24), the ratio of […]_{|N|} elliptic shifted factorials gives (after simplifications) −2|N|βγ. The product of ratios of […]_{N_{j}} yields ∑_{j=1}^{n} N_{j}((1 + g_{j})^{2} + (1 + g_{j} − β − γ)^{2} − (1 + g_{j} − β)^{2} − (1 + g_{j} − γ)^{2}), which is seen to cancel the previous expression. This proves the claimed modularity property.

Turn now to the D_{n} sum (22). Ratios of the series coefficients in the left-hand side look as follows

h_{ℓ}(λ) = q^{ℓ} ∏_{{j=1 \atop j≠ℓ}}^{n} \frac{θ(t_{j}q^{λ_{j}+|λ|+1}, bcdt_{j}^{−1}q^{|λ|−λ_{j}}; p)}{θ(t_{j}q^{λ_{j}+|λ|}, t_{ℓ}t_{j}q^{λ_{j}+λ_{ℓ}+1}/bcd; p)} \frac{θ(t_{ℓ}q^{λ_{ℓ}+|λ|+2}; p)}{θ(t_{ℓ}q^{λ_{ℓ}+|λ|}; p)}

× ∏_{i=1}^{ℓ−1} \frac{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}−1}; p)}{θ(t_{i}t_{ℓ}^{−1}q^{λ_{i}−λ_{ℓ}}; p)} ∏_{j=ℓ+1}^{n} \frac{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}+1}; p)}{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}−λ_{j}}; p)}

× ∏_{j=1}^{n} \frac{θ(t_{ℓ}t_{j}q^{1+N_{j}+λ_{ℓ}}/bcd, t_{ℓ}t_{j}^{−1}q^{−N_{j}+λ_{ℓ}}, t_{j}q^{|λ|}; p)}{θ(t_{ℓ}t_{j}^{−1}q^{λ_{ℓ}+1}, t_{j}q^{1+N_{j}+|λ|}, bcdt_{j}^{−1}q^{|λ|−N_{j}}; p)}

×\frac{θ(bq^{|λ|}, cq^{|λ|}, dq^{|λ|}; p)}{θ(t_{ℓ}q^{λ_{ℓ}+1}/b, t_{ℓ}q^{λ_{ℓ}+1}/c, t_{ℓ}q^{λ_{ℓ}+1}/d; p)}.  (25)

The shift q^{λ_{k}} → pq^{λ_{k}} does not change these functions, that is, they are elliptic in the λ_{k} variables. For k < ℓ or k > ℓ, it follows from the cancellation of quasiperiodicity multipliers

q^{−n}, −\frac{t_{k}t_{ℓ}}{bcd} q^{1+λ_{k}+λ_{ℓ}}, ∏_{{j=1 \atop j≠k,ℓ}} (−\frac{t_{j}}{bcd}q^{λ_{j}−|λ|}) ,

appearing from the product ∏_{j=1,j≠ℓ}^{n}, the factor q^{−2} coming from the next ratio of theta functions, q (from the next two products), and ∏_{j=1}^{n}(−bcdt_{j}^{−1}q^{|λ|+1}), −q^{−3|λ|}/bcd (from the last two lines expressions, respectively). For k = ℓ we have multipliers q^{1−n} (from the first ratio), ∏_{j=1,j≠ℓ}^{n}(t_{j}^{2}t_{ℓ}q^{2λ_{j}+λ_{ℓ}−|λ|+1}/b^{2}c^{2}d^{2}) (from the second ratio), q^{−4} (third ratio), q^{1−n} (next two products), and

∏_{j=1}^{n} \frac{b^{2}c^{2}d^{2}}{t_{ℓ}t_{j}^{2}} q^{1+|λ|−λ_{ℓ}}, \frac{t_{ℓ}^{3}}{b^{2}c^{2}d^{2}} q^{3(1+λ_{ℓ}−|λ|)}

(from the last two lines, respectively). Their product equals 1, that is, the functions h_{ℓ} are elliptic in all λ_{k}.

After rewriting sum (22) in terms of the elliptic numbers

∑_{{0≤λ_{j} ≤N_{j} \atop j=1,... ,N_{j}}} ∏_{j=1}^{n} \frac{[g_{j} + λ_{j} + |λ|]}{[g_{j}]} ∏_{1≤i<j≤n} \frac{[g_{i} − g_{j} + λ_{i} − λ_{j}]}{[g_{i} − g_{j}]}

×\frac{∏_{i,j=1}^{n}[g_{i} + g_{j} + 1 + N_{j} − β − γ − δ]_{λ_{i}}}{∏_{1≤i<j≤n}[g_{i} + g_{j} + 1 − β − γ − δ]_{λ_{i}+λ_{j}}}

× ∏_{i,j=1}^{n} \frac{[g_{i} − g_{j} − N_{j}]_{λ_{i}}}{[1 + g_{i} − g_{j}]_{λ_{i}}} ∏_{j=1}^{n} \frac{[g_{j}]_{|λ|}[β + γ + δ − g_{j}]_{|λ|−λ_{j}}}{[g_{j} + 1 + N_{j}, β + γ + δ − g_{j} − N_{j}]_{|λ|}}

×\frac{[β, γ, δ]_{|λ|}}{∏_{j=1}^{n}[g_{j} + 1 − β, g_{j} + 1 − γ, g_{j} + 1 − δ]_{λ_{j}}}

= ∏_{j=1}^{n} \frac{[g_{j} + 1, g_{j} + 1 − β − γ, g_{j} + 1 − β − δ, g_{j} + 1 − γ − δ]_{N_{j}}}{[g_{j} + 1 − β − γ − δ, g_{j} + 1 − β, g_{j} + 1 − γ, g_{j} + 1 − δ]_{N_{j}}} , (26)

permutational symmetry between (λ_{k}, g_{k}, N_{k}) and (λ_{m}, g_{m}, N_{m}), for fixed k and m, becomes evident.

Ellipticity in the variables g_{k}, k ≠ ℓ, follows from the cancellation of the following sequence of multipliers in (25), appearing after the shift t_{k} → pt_{k}: q^{−1}, −t_{k}t_{ℓ}q^{1+λ_{k}+λ_{ℓ}}/bcd, −p^{−1}bcdq^{|λ|−λ_{k}}t_{k}^{−1}, q, pq^{−1−|λ|−λ_{ℓ}}t_{ℓ}^{−1}. Similarly, for g_{ℓ} we get the multipliers (in the order of appearance) ∏_{j=1,j≠ℓ}^{n}(−t_{j}t_{ℓ}q^{1+λ_{j}+λ_{ℓ}}/bcd), q^{−2}, q^{1−n}, ∏_{j=1,j≠ℓ}^{n}(−bcdq^{−λ_{ℓ}}/t_{j}t_{ℓ}), −bcdq^{−1−|λ|−2λ_{ℓ}}/t_{ℓ}^{3}, −t_{ℓ}^{3}q^{3+3λ_{ℓ}}/bcd, which total in 1. Ellipticity in β (or γ, δ) is confirmed by analogous calculation, which we skip for brevity.

The proof of the modularity of h_{ℓ} requires essentially longer computations. Skipping the original expression for the difference of squares of arguments of elliptic numbers in h_{ℓ}, the first chain of obvious simplifications in it results in the sum

2(1 + |λ| + g_{ℓ} + λ_{ℓ})(λ_{ℓ} − |λ| − N_{ℓ} + 1) + (2g_{ℓ} − β − γ − δ + N_{ℓ} + 1 + λ_{ℓ})^{2}

−(g_{ℓ} − β − γ − δ + N_{ℓ} − |λ|)^{2} + ∑_{ρ=β,γ,δ} ((ρ + |λ|)^{2} − (g_{ℓ} + λ_{ℓ} + 1 − ρ)^{2}) ,

which is equal to zero.

It is more easy to work with the right-hand side of (22). Its ellipticity in parameters follows from the fact that the ratio

\frac{[g_{j}, g_{j} − β − γ, g_{j} − β − δ, g_{j} − γ − δ]}{[g_{j} − β − γ − δ, g_{j} − β, g_{j} − γ, g_{j} − δ]}

is elliptic in all variables. The modular invariance is guaranteed by the following equality

g_{j}^{2} + (g_{j} − β − γ)^{2} + (g_{j} − β − δ)^{2} + (g_{j} − γ − δ)^{2}
= (g_{j} − β − γ − δ)^{2} + (g_{j} − β)^{2} + (g_{j} − γ)^{2} + (g_{j} − δ)^{2}.

Thus, we have proved total ellipticity and modularity of both sides of both A_{n} and D_{n} sums.

Corollary 5. The Warnaar C_{n}, elliptic Milne A_{n}, and elliptic Bhatnagar-Schlos-ser D_{n} summation formulae (18), (21), and (22), respectively, are valid in the small σ expansion up to the order σ^{12}.

Proof. For a sufficiently small absolute value of the parameter σ, both sides of the equalities (18), (21), (22) are holomorphic on the upper half plane of the modular parameter τ, Im(τ) > 0. Being P SL(2, ℤ) invariant, they have σ-expansions of the form ∑_{k=0}^{∞} m_{k}(τ)σ^{2k}, where m_{k}(τ) are modular forms of the weight 2k. For Im(τ) → ∞ (i.e. p → 0), the differences of expressions on the left-hand and right-hand sides of these equalities vanish. Therefore in the expansions of these differences in σ, only cusp forms may appear. Since there are no modular cusp forms of the weight smaller than 12, the C_{n}, A_{n}, D_{n} summation formulae under consideration are valid in the σ-expansion up to the terms of the order σ^{12}. These arguments are repetive to the ones, which were used already in [6, 7].

In addition to providing substantial evidence in favor of the main conjecture stated at the beginning, the results of this note show that the properties of total ellipticity and modular invariance give a quite simple tool for conjecturing elliptic generalizations of certain basic hypergeometric series identities. The complete proofs of the C_{n} sum of [7] and A_{n}, D_{n} sums (21), (22), reached by Rosengren in [16], strengthen the latter statement. It is natural to pose here a question on the existence of similar identities for other root systems.

We would like to conclude by a remark that the approach of [20] to elliptic hypergeometric series can be generalized to some extent by employing multidimensional Riemann theta functions. For this, it is necessary to demand that the functions h_{ℓ}(λ) in (2) are Abelian functions (i.e. multiple periodic functions living on a gdimensional complex torus) or rational functions on a compact Riemann surface of arbitrary genus. We hope to consider elsewhere these possibilities in detail.

The author would like to acknowledge useful discussions with G.E. Andrews, A. Berkovich, J.F. van Diejen of possible consequences of the elliptic Bailey chain and to thank H. Rosengren for confirming, prior to publication [16], that he proved the conjectured sums (21), (22). This work has been done at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality are greatfully acknowledged. It was partially supported also by the Russian Foundation for Basic Research (RFBR) grant No. 00-01-00299.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany and Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia