Gustafson-Rakha-Type Elliptic Hypergeometric Series

We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.


Introduction
By combining two integral evaluations previously obtained by Gustafson [6], Gustafson and Rakha [7] evaluated the basic hypergeometric integral the present paper is to give a direct proof of Spiridonov's conjectured summation and to apply it to derive some further summation and transformation formulas. It is worth mentioning that Spiridonov's elliptic extension of (1.1) can be interpreted as the identity between superconformal indices of two dual quantum field theories [17,. This indicates that (1.1) and related results are not mere curiosities and that it is not unreasonable to expect further applications.
The plan of the paper is as follows. In Section 3, we prove Spiridonov's conjecture. The proof is elementary and provides in particular a significant simplification of the trigonometric case. The only previously known proof of the Gustafson-Rakha summation is the original one, which as we recall is based on first proving two auxiliary multiple integral evaluations, combining them to obtain (1.1) and finally on a technical computation to pass from integrals to finite residue sums. In Section 4, we give some applications of our result. Namely, combining the elliptic Gustafson-Rakha sum with a summation from [14], we obtain two transformation formulas and two further summation formulas for multivariable elliptic hypergeometric series. These four results are all new even in the trigonometric case.
Note added in proof: After completing this work, I learned from Masahiko Ito and Masatoshi Noumi that they have independently proved Theorem 3.1, using a different method.

The elliptic Gustafson-Rakha summation
Our main result is the following identity. It is easy to see that the case p = 0 is equivalent to [7, Theorem 1.2] and the general case to the conjecture of [16, p. 953]. Recall the notation Z = z 1 · · · z n .
Theorem 3.1. For parameters subject to q N −1 b 1 · · · b 4 z 2 1 · · · z 2 n = 1, (3.1) We will prove Theorem 3.1 by induction on N . In the case N = 1, we have x i = δ ik for some k. Using k as summation index, Theorem 3.1 reduces to the following theta function identity.
Proof . We apply induction on n, starting from the trivial case n = 1. Let b 1 = vw and b 2 = v/w, with v and w free parameters. As a function of w, each term in the sum, as well as the right-hand side, has the form f (w) = Cθ(aw, a/w) with C and a independent of w. It is a classical fact that any such function is determined by its values at two generic points. Indeed, Weierstrass' identity (which is equivalent to the case n = 2 of (2.2)) states that provided that bc, b/c / ∈ p Z . Thus, it suffices to verify (3.2) for two independent values of b 1 . Assuming n ≥ 2, we choose b 1 = 1/z n−1 and b 1 = 1/z n . By symmetry, it is enough to consider the second case. Then, the term corresponding to k = n cancels and we are reduced to an identity equivalent to (3.2), with n replaced by n − 1 and b 1 by z n .
We mention that it is not hard to deduce (3.2) from classical theta function identities. Indeed, let t = b n+1 in (2.2) (so that the right-hand side vanishes) and then make the substitutions n → n + 4, (z 1 , . . . , z n ) → (z 1 , . . . , z n , 1, −1, one may deduce that the left-hand side of (3.2) is equal to The fact that this equals the right-hand side of (3.2) follows from Jacobi's fundamental formulae [21,Section 21.22]. The inductive step in the proof of Theorem 3.1 is almost identical to that of [12, Theorem 5.1]. Denoting the right-hand side of (3.1) by R N (Z; b 1 , b 2 , b 3 , b 4 ) (where we for a moment consider Z as a free variable) we observe that, regardless of the parity of n, Assuming (3.1) for fixed N , it follows that where Z = z 1 · · · z n and q N BZ 2 = 1. We pull the factor R 1 inside the sum and expand it using (3.1), with z i replaced by q x i z i . This gives where we replaced each x i by x i − y i and used that y i y j = 0 for i = j. By elementary manipulations, using the expression above can be rewritten Writing y i = δ ik , the inner sum takes the form . By (2.2), this can be evaluated as and we arrive at (3.1) with N replaced by N + 1. This completes the proof of Theorem 3.1.
We will now rewrite (3.1) in a way that hides some of its symmetry but makes it clear that it generalizes the Frenkel-Turaev summation (2.3). To this end, we replace n by n + 1, z n+1 by q −N a −1 and eliminate x n+1 from the summation. After routine simplification, we arrive at the following identity.

Applications
The elliptic Bailey transformation (2.4) can be derived from the elliptic Jackson summation (2.3). Similar arguments can be used in multivariable situations, see, e.g., [1,2,8] for the trigonometric and [12] for the elliptic case. We will use this method to derive a new multivariable elliptic Bailey transformation by combining the two multivariable elliptic Jackson summations (2.5) and (3.3).
When n is odd, the value of this sum can be rewritten which leads to the right-hand side of (4.1). The case of even n is treated similarly.
One may obtain further transformation formulas by iterating Theorem 4.1. We will only give one example, exploiting the fact that the left-hand side of (4.1) is invariant under interchanging c and e. In the identity expressing the corresponding symmetry of the right-hand side, we make the substitutions (λ, a, b, c, d) → (a, a 2 q/bcd, aq/cd, aq/bd, aq/bc), keeping e, f, g, z 1 , . . . , z n fixed. This leads to another multivariable elliptic Bailey transformation.
Theorem 4.1 reduces to Corollary 3.3 when aq = bc. More interestingly, when b = 1 the left-hand side of (4.1) reduces to 1. After a change of parameters, this leads to the following new multivariable elliptic Jackson summation. Corollary 4.3. If a 2 q N +1 = bcdez 2 1 · · · z 2 n , then with the special case aq = bc of (2.5), when the right-hand side is equal to valid for a 2 q N +1 = bcdez 2 1 · · · z 2 n and t arbitrary. This identity is less novel than Corollary 4.3, as it can be deduced from Theorem 3.1 in a more direct manner. Indeed, writing the sum as N k=0 x 1 ,...,xn≥0 x 1 +···+xn=k (· · · ), the inner sum is computed by Theorem 3.1 and the outer sum by (2.3). In fact, the same proof gives the following more general result, which reduces to (4.2) when (d, e) = (f Z, gZ) or (Z, f gZ) if n is odd or even, respectively.