Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation

We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.


Introduction
Previously, one of us [21] established an elliptic Taylor expansion theorem which extends Ismail's [11] expansion for functions symmetric in z and 1/z in terms of the Askey-Wilson monomial basis. The expansion theorem in [21] involves a special case of Rains' [17] elliptic extension of the Askey-Wilson divided difference operator. As applications, new simple proofs were given for Frenkel and Turaev's [8] elliptic extensions of Jackson's 8 φ 7 summation and of Bailey's 10 φ 9 transformation. A further application concerned the computation of the connection coefficients of Spiridonov's [23] elliptic extension of Rahman's biorthogonal rational functions.
Here we take a closer look at elliptic Taylor expansions. In particular, we describe the action of the m-th elliptic divided difference on a function, expressed in terms of the function. In the ordinary case, if δ h denotes the central difference operator, defined by , the m-th difference is given by For the q-case, where δ h is replaced by the Askey-Wilson operator D q , acting on functions f (z) symmetric in z and 1/z, an explicit formula for D m q f (z) was established by Cooper [5]. One of the results of our paper concerns an extension of Cooper's formula to the elliptic setting. We remark that Ismail, Rains and Stanton [12] independently have also proved an elliptic extension of Cooper's formula which turns out to be equivalent to our result by a multiplication of operators. In [14], Ismail and Stanton have used Cooper's explicit formula to work out an explicit interpolation formula for polynomials symmetric in z and 1/z. Likewise, we use our elliptic extension of Cooper's formula to find an elliptic interpolation formula. Application of this formula yields single and multivariable identities of Karlsson-Minton type.
This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html Ismail and Stanton [13] not only considered Taylor expansions in terms of the Askey-Wilson monomial basis {(az, a/z; q) n , n ≥ 0} (see the subsequent subsection for the q-shifted factorial notation), but also in terms of the basis q 1 4 z, q 1 4 /z; q 1 2 n , n ≥ 0 , for which they deduced quadratic summations as applications. We are able to extend Ismail and Stanton's analysis and provide, in particular, a Taylor expansion for an elliptic extension of this other basis. We note that in addition, Ismail and Stanton [13, Theorem 2.2] gave a Taylor expansion theorem for the basis (1 + z 2 ) −q 2−n z 2 ; q 2 n−1 z −n , n ≥ 0 , however this result (which involves an evaluation at z = 0) appears not to extend to the elliptic setting.
Finally, we consider series partially involving products of S. Bhargava's [3] cubic theta functions. Such series have not been considered before. We introduce two different cubic theta extensions of shifted factorials which are designed in such forms that they behave well under the iterated action of the elliptic Askey-Wilson operator. Applications of Taylor expansion yield cubic theta extensions of Jackson's 8 φ 7 summation formula and of a quadratic summation of Gessel and Stanton.
Before we present our new results, to make this paper more self-contained, we briefly review some important material from the theory of elliptic hypergeometric series. Afterwards we turn to the Askey-Wilson operator and its elliptic extension, and then we provide our new results.

Elliptic hypergeometric series
For basic hypergeometric series, see Gasper and Rahman's textbook [9]. Elliptic hypergeometric series are treated there in Chapter 11.
By definition, a function is elliptic if it is meromorphic and doubly periodic. It is well known (cf., e.g., [25]) that elliptic functions can be built from quotients of theta functions.
As building blocks we will use the modified Jacobi theta function with argument x and nome p, defined (in multiplicative notation) by where x, x 1 , . . . , x m = 0, |p| < 1. The modified Jacobi theta functions satisfy the following basic properties which are essential in the theory of elliptic hypergeometric series: (a k ; q, p) n , for compact notation. For p = 0 we have θ(x; 0) = 1 − x and, hence, (a; q, 0) n = (a; q) n is a q-shifted factorial in base q. The parameters q and p in (a; q, p) n are called the base and nome, respectively. Observe that (pa; q, p) n = (−1) n a −n q −( n 2 ) (a; q, p) n , which follows from (1.1b). A list of other useful identities for manipulating the q, p-shifted factorials is given in [9,Section 11.2]. A series c n is called an elliptic hypergeometric series if g(n) = c n+1 /c n is an elliptic function of n with n considered as a complex variable, i.e., the function g(x) is a doubly periodic meromorphic function of the complex variable x. Without loss of generality, by the theory of theta functions, one may assume that where the elliptic balancing condition, namely holds. If we write q = e 2πiσ , p = e 2πiτ , with complex σ, τ , then g(x) is indeed periodic in x with periods σ −1 and τ σ −1 .
For convergence reasons, one usually requires a s+1 = q −n (n being a nonnegative integer), so that the sum of an elliptic hypergeometric series is in fact finite.

The Askey-Wilson operator
The Askey-Wilson operator D q was first defined in [1]. We consider meromorphic functions f (z) symmetric in z and 1/z. Writing z = e iθ (note that θ need not to be real), we may consider f to be a function in x = cos θ = (z + 1/z)/2 and write f [x] := f (z). (I.e., f can be considered as a function in z, or equivalently, as a function in x, where the two different notations specify the dependency to be considered.) The Askey-Wilson operator acts on functions of x = cos θ. It is defined as follows: where ι[x] = x (i.e., ι(z) = (z + 1/z)/2). Equation (1.4) can also be written as The operator D q is a q-analogue of the differentiation operator (which is different to Jackson's q-difference operator). In particular, since where T n [cos θ] = cos nθ and U n [cos θ] = sin(n + 1)θ/ sin θ are the Chebyshev polynomials of the first and second kind, one easily sees that D q maps polynomials to polynomials, lowering the degree by one.
In the calculus of the Askey-Wilson operator the so-called "Askey-Wilson monomials" φ n (x; a) = (az, a/z; q) n form a natural basis for polynomials or power series in x. One readily computes Ismail [11] proved the following Taylor theorem for polynomials f [x].
Ismail and Stanton [13] extended the above polynomial Taylor theorem to hold for entire functions of exponential growth, resulting in infinite Taylor expansions. Marco and Parcet [15] extended this yet further to hold for arbitrary q-differentiable functions, resulting in infinite Taylor expansions with explicit remainder term. Among other results they were able to recover the nonterminating q-Pfaff-Saalschütz summation (cf. [9, Appendix (II.24)]).
we were led in [21] to define a c-generalized well-poised Askey-Wilson operator acting on x (or z) by which acts "degree-lowering" on the "rational monomials" (or "well-poised monomials") (az, a/z; q) n (cz, c/z; q) n in the form Clearly, D 0,q = D q . More generally, for parameters c, q, p with |q|, |p| < 1, we defined an elliptic extension of the Askey-Wilson operator, acting on functions symmetric in z ±1 , by Note that D c,q,0 = D c,q . In particular, using (1.1c), we have D c,q,p (az, a/z; q, p) n (cz, c/z; q, p) n = (−1)2aθ c/a, acq n−1 , q n ; p θ(q; p) aq 1 2 z, aq (1.6) Remark 1.2. The operator D c,q,p happens to be a special case of a multivariable difference operator introduced by Rains in [16]. Already in the single variable case Rains' operator involves two more parameters than D c,q,p . (Rains' difference operators generate a representation of the Sklyanin algebra, as observed in [16] and made explicit in [18] and [19,Section 6].) Rains' operator can be specialized to act as degree-lowering (as the above D c,q,p does), degree-preserving or degree-raising on abelian functions. Rains used his multivariable difference operators in [16] to construct BC n -symmetric biorthogonal abelian functions which generalize Koornwinder's orthogonal polynomials. He further used his operator in [17] to derive BC n -symmetric extensions of Frenkel and Turaev's 10 V 9 summation and 12 V 11 transformation.

Elliptic Taylor expansions and interpolation
We work in the following space of abelian functions. For a complex number c, let where g k (z) runs over all functions being holomorphic for z = 0 with g k (z) = g k (1/z) and In classical terminology, g k (z) is an even theta function of order 2k and zero characteristic. Rains [17] refers to such functions as BC 1 theta functions of degree k, whereas in Rosengren and Schlosser [20] they are referred to as D k theta functions. It is well-known that the space V k of even theta functions of order 2k and zero characteristic has dimension k +1 (see, e.g., Weber [25, p. 49]).
Note that W n c consists of certain abelian functions. (For p → 0 these degenerate to certain rational functions which we may call "well-poised".) Lemma 2.1 ([21, Lemma 4.1]). For any arbitrary but fixed complex number a (satisfying a = cq j p k , for j = 0, . . . , n − 1, and k ∈ Z, and a = q j p k /c, for j = 2 − 2n, . . . , 1 − n, and k ∈ Z), the set (az, a/z; q, p) k (cz, c/z; q, p) k , 0 ≤ k ≤ n forms a basis for W n c .
Note that, in view of (1.6), the elliptic Askey-Wilson operator maps functions in W n c to functions in W n−1 cq 3 2 . We now define c,q,p = ε, the identity operator. We have the following elliptic Taylor expansion theorem which extends Theorem 1.1 of Ismail.
We now prove an elliptic extension of a theorem of S. Cooper [5] which explicitly describes the action of the m-iterated Askey-Wilson operator. c,q,p on a function f ∈ W n c is given by Proof . We prove this by induction. If m = 1, then (2.1) just reduces to the definition of Hence the theorem is proved. Notice that in the last step we used the addition formula (1.1c) with the substitutions to simplify the summand. (which acts on functions h(z) = g(z)/(czq m−1 , cq m−1 /z; q, p) n−m+1 with g(z) = g(1/z) and g(pz) = p −n z −2n g(z)) is independent of c, thus our operator D We are now able to obtain an elliptic extension of Ismail and Stanton's [14, Theorem 3.4] interpolation formula. In particular, for any a ∈ C, the function f ∈ W n c is uniquely determined by its evaluation at the n + 1 interpolation points a, aq, . . . , aq n , with closed form coefficients.
The last sum was obtained by virtue of the Frenkel and Turaev summation formula (1.2). The theorem then follows by elementary manipulations.
More generally, we have the following result.
Proof . We take and apply Theorem 2.6.

Remark 2.9.
It should be noted that if in the proof of Corollary 2.8 we instead would have taken for 0 ≤ t ≤ n, we would have just obtained the special case of Corollary 2.8 with b j → cq j−1 for t + 1 ≤ n, which is clear carrying out those specializations in (2.2).
We define a multivariate extension of the elliptic Askey-Wilson operator as follows.
We combine Theorems 2.10 and 2.11 to obtain the following multivariable elliptic interpolation formula.
This theorem extends a result given by Ismail and Stanton [14, Theorem 3.10], which can be obtained by taking m = 2, p → 0, c 1 = c 2 = 0 and n 1 = n 2 = n.
Corollary 2.13. We have the following multivariable elliptic Karlsson-Minton type identity Proof . We apply Theorem 2.12 to for i = 1, . . . , m.
Corollary 2.13 extends a result by Ismail and Stanton (see [14,Corollary 3.11]), corresponding to a special case of its m = 2 instance.
More generally, f (z 1 , . . . , z m ) could involve symmetrized products of 2 k factors of the form (λz ± i 1 z ± i 2 · · · z ± i k ; q, p) y (the notation z ± i j means that the resprective variable could appear as z i j or z −1 i j , where all possible combinations appear), where {i 1 , . . . , i k } is any subset of {1, . . . , n}. (In the corollary, we only considered factors for k = 1, 2.) Corollary 2.13 can be easily seen to be equivalent to its u l ij = 1 and v ij = 1 case, for all i, j, in which case the respective factorials reduce to simple theta functions. To recover the general case from this special case one can suitably increase r ij and s 1 , . . . , s m and choose the parameters partially in geometric progression to obtain shifted factorials. In particular, we can replace r ij by u 1 + · · · + u r ij and relabel α u 1 +···+u l ij −1 +h → α l ij q h−1 , for all 1 ≤ l ij ≤ r ij , 1 ≤ h ≤ u l ij , etc. (One could even add extra bases, in addition to q. This feature is typical for series of Karlsson-Minton type.) For convenience, we restate the corollary in this equivalent form.
Corollary 2.14. We have the following multivariable elliptic Karlsson-Minton type identity θ(α l ij z i z j , α l ij z i /z j , α l ij z j /z i , α l ij /z i z j ; p) θ(α l ij a i a j , α l ij a i /a j , α l ij a j /a i , α l ij /a i a j ; p)   = n 1 ,...,nm q −2k i θ α l ij a i a j q k i +k j , q k i +k j a i a j /α l ij , α l ij a i q k i −k j /a j , q k i −k j a i /a j α l ij ; p θ(α l ij a i a j , a i a j /α l ij , α l ij a i /a j , a i /a j α l ij ; p) for i = 1, . . . , m.

Expansions involving cubic theta functions
The cubic theta function γ(z, a; p) with two independent variables z and a in addition to the nome p was considered by S. Bhargava [3]. (For a thorough treatment of the theory of cubic theta functions in analogy to the theory of the classical Jacobi theta functions, see [22].) It is defined by This function, up to a normalization factor p 2 ; p 2 2 ∞ (independent from a and z), is almost equal to the following product of two modified Jacobi theta functions which differs by the factor p kl to the summand of the double series in (3.1). Because of this additional factor p kl , the cubic theta function does not factorize into a product of two modified Jacobi theta functions of such a simple form. In principle though, the cubic theta function could be factorized into two modified Jacobi theta functions, but their arguments would have nontrivial expansions in a, z, and p. From (3.1), by replacing (k, l) by (l, k), or (k, l) by (−l, −k), respectively, we immediately deduce the symmetries [3] γ(1/z, a; p) = γ(z, a; p), (3.2a) and γ(z, 1/a; p) = γ(z, a; p). (3.2b) Further, from (3.1), by replacing (k, l) by (k + λ + µ, l + λ), it is easy to verify that for all integers λ and µ the following functional equation holds [3]: γ(z, a; p) = p 3λ 2 +3λµ+µ 2 a 2λ+µ z µ γ p µ/2 z, p 3(2λ+µ)/2 a; p .
Cooper and Toh [6] proved the following addition formulae which will be useful in our computations.
Lemma 3.1 ([6, Corollary 4.5]). The following identities connecting modified Jacobi theta functions and cubic theta functions hold: These two identities were proved in [6] by specializing a (3 × 3) determinant evaluation involving cubic theta functions. They can also be proved directly, expanding the cubic theta functions and modified Jacobi theta functions as infinite series, together with clever series rearrangement. Now we introduce the first cubic theta analogue of the q-shifted factorial by az, a/z; q, p n := From (3.3a) it is easy to see that the cubic shifted factorial satisfies apz, a/pz; q, p n = 1 p n z 2n az, a/z; q, p n .
Together with (3.2a), this implies that the quotient az, a/z; q, p n (cz, c/z; q, p) n is in the space W n c . Hence we can apply Theorem 2.2 to it, by which we obtain the first cubic theta extension of Jackson's 8 φ 7 summation (1.3).
To recover Jackson's 8 φ 7 summation from Corollary 3.2, substitute a → −a p(1 + a 2 q n−1 ) in (3.5), multiply both sides of the identity by (1 + a 2 q n−1 ) n and let p → 0. When p → 0, the usual theta shifted factorials clearly reduce to the q-shifted factorials. That is, the quotient on the left-hand side reduces to lim p→0 (bc, c/b; q, p) n (cz, c/z; q, p) n = (bc, c/b; q) n (cz, c/z; q) n .
We take similar limits on the right-hand side of (3.5). We take similar limits on the right-hand side of equation (