Multidimensional matrix inversions and elliptic hypergeometric series on root systems

Multidimensional matrix inversions is a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As applications, we obtain a new $A_r$ elliptic Jackson summation, as well as several quadratic, cubic and quartic summation formulas.


Introduction
Explicit matrix inversions provide a powerful tool in the study of special functions. In particular, they have been used to derive quadratic, cubic and quartic summations for one-variable basic hypergeometric series, see e.g. [18,19,20,33,34]. Moreover, Andrews [1] found that the Bailey transform, which is useful for deriving Rogers-Ramanujan-type identities, is closely related to a certain matrix inversion. A unified generalization of several inversions used in these contexts was obtained by Krattenthaler [23]. He proved that the lower-triangular matrices (f nk ) n,k∈Z and (g kl ) k,l∈Z with entries are mutually inverse. The goal of the present work is to obtain generalizations of (1.1) that are both multi-dimensional and elliptic, so that we can apply them to multiple elliptic hypergeometric series. Although elliptic hypergeometric series first appeared in the 1980's [13], they did not take off as a mathematical research area until after the seminal paper of Frenkel and Turaev in 1997 [15]. In one of the first papers on this subject [53,Lemma 3.2], Warnaar gave an elliptic extension of Krattenthaler's matrix inverse and applied it to obtain elliptic analogues of many identities from [18,19,20,34].
Matrix inversion techniques have also been applied to multiple hypergeometric series. One then needs multi-dimensional matrices, which in the cases of relevance to us have rows and columns labelled by Z r . In the 1990's, Milne and coworkers found several multi-dimensional Bailey transforms and matrix inversions, which were applied to basic hypergeometric series associated with root systems [4,28,29,30,31]. The second author [45,Thm. 3.1] gave a unified extension of these inversions, which reduces to (1.1) in the one-variable case. This allowed him to obtain quadratic and cubic summation formulas for multiple hypergeometric series; see [24,25,26,46,47,48,49,50] for subsequent work in this direction and [51] for a general overview of classical and basic hypergeometric series associated with root systems.
There has been a substantial amount of work on elliptic hypergeometric series associated with root systems, see [44] and references given there. In particular, Bhatnagar and the second author [5] obtained several elliptic Bailey transforms associated with root systems, and also gave the corresponding multidimensional matrix inversions explicitly. However, until now these have not been extended to the level of generality of (1.1). In the present paper we address this question.
In §3 we give two multidimensional extensions of Warnaar's matrix inversion, as well as one somewhat less general inversion. In §4 we give applications to summation formulas for elliptic hypergeometric series on root systems. More precisely, in §4.2 we obtain a new extension of Jackson's summation and two quadratic summations, in all three cases for series associated with the root system A r . In §4.3 we obtain a quadratic, a cubic and a quartic summation for series traditionally associated with the root system D r . For all these summations except the last one, we also give companion identities, where the summation is over a simplex rather than a hyper-rectangle. One of the quadratic A r summations (Theorem 4.3) and the quartic D r summation (Theorem 4.13) are new even in the limit case of basic hypergeometric series.
The multiple hypergeometric series appearing in this paper are of what different authors have called "type I", "Dixon-type" or "Gustafson-Milnetype". For so called "type II" or "Selberg-type" series, a matrix inversion was found independently by Coskun and Gustafson [12,Eq. (4.16)] and Rains [35,Cor. 4.3], see also [11] for the corresponding Bailey transform. It is not clear whether this inversion can be extended to the generality of (1.1). A quadratic summation for Selberg-type series is given in [37,Cor. 4.3] and further results are forthcoming [27]. There are also related integral identities [9,36,37,38]. To our knowledge, the present paper contains the first known quadratic summations for multiple elliptic hypergeometric series of type I, and the first known cubic and quartic summations for either type.
There is a close correspondence between elliptic hypergeometric functions given by finite sums and by integrals [44]. In particular, Spiridonov and Warnaar [52] gave several multiple integral inversions, which can be viewed as continuous analogues of matrix inversions from [5]. These integral inversions and their associated Bailey transforms have found a role in quantum field theory, see e.g. [6,7,8,16,17,32,54]. This may serve as a motivation to look for integral analogues of some of our results.
Finally, we mention that it seems possible to generalize many of our results to transformation formulas. This will be the subject of future work.

Preliminaries
2.1. Theta functions. We recall some classical results for theta functions. We will write Here, p is a complex number with |p| < 1, which will be fixed throughout and suppressed from the notation. We refer to the case p = 0, θ(x) = 1 − x as the trigonometric case. We will sometimes use the shorthand notation θ(x 1 , . . . , x n ) = θ(x 1 ) · · · θ(x n ), θ(x 1 y ± , . . . , x n y ± ) = θ(x 1 y)θ(x 1 /y) · · · θ(x n y)θ(x n /y). Among the properties of θ we mention the inversion formula and Weierstrass' addition formula We will also need the identity To our knowledge, it was first obtained by Gustafson [21,Lemma 4.14], see [39] for further references and comments. We will use the following terminology.
Definition 2.1. A theta function of degree k, nome p and norm t is a holomorphic function on C \ {0} such that It is a classical fact (see e.g. [42,Cor. 1.3.5]) that any function satisfying these conditions can be written as In particular, the zero set of f is the union of the geometric progressions a j p Z . One consequence is the following classical result.
Lemma 2.2. Let f be a theta function of degree k, nome p and norm t and assume that where b j are non-zero complex numbers such that Then, f is identically zero.

Elliptic hypergeometric series.
A one-variable elliptic hypergeometric series k a k is a formal or convergent series such that a k+1 /a k = f (k) for some elliptic function f . The standard notation for such series is based on elliptic shifted factorials As before, the parameter p will be fixed and suppressed from the notation. The parameter q will only be written out when different values of q appear in the same identity. We will use the condensed notation (a 1 , . . . , a m ) k = (a 1 ) k · · · (a m ) k .
Throughout the paper we implicitly assume that all parameters are generic, so that we never divide by zero. In particular, it is convenient to assume that q k / ∈ p Z for k ∈ Z ≥0 . Otherwise the factor (q; q) k , which frequently appears in denominators, vanishes.
We will write Elliptic hypergeometric series associated with the root system A r are multiple series, which apart from elliptic shifted factorials contain the characteristic where k j are summation indices. Rational limit cases of such series play a role in the representation theory of unitary groups [22]. We will often write |k| = i k i . We will need the A r elliptic Jackson summation where a 2 q |n|+1 = bcde. It is due to the first author [39,Cor. 5.3] in general and to Milne [29] when p = 0. We will also need the C r elliptic Jackson summation where a 2 q |n|+1 = bcde. It is due to the first author [39,Cor. 5.3] in general and, independently, to Denis and Gustafson [14] and Milne and Lilly [31] when p = 0.

Elliptic multidimensional matrix inversions
We will consider matrices f = (f nk ) n, k∈Z r labelled by pairs of multiindices. All matrices that appear are lower-triangular in the sense that f nk = 0 unless n ≥ k, that is, n i ≥ k i for all i. Then, f and g = (g nk ) n, k∈Z r are inverse matrices if l≤k≤n f nk g kl = δ nl (3.1) or, equivalently, Our first main result is the following explicit pair of inverse matrices. Since it will be used to obtain results for multiple hypergeometric series associated with the root system A r , we refer to it as an elliptic A r matrix inversion.
As we explain in more detail in §3.1, Theorem 3.1 contains two matrix inversions due to Bhatnagar and the second author [5] as special cases. The case p = 0 of Theorem 3.1 is [45, Thm. 3.1]. (The additional parameter b present in [45] may be suppressed by a change of variables.) It contains several previously known matrix inversions [4,28,29,30] as special cases. The case r = 1 is due to Warnaar [53, Lemma 3.2], see also [43].
We will prove Theorem 3.1 by verifying (3.1). We assume n > l, since otherwise (3.1) is trivial. Pulling out the factors independent of k, we then have to prove that This identity can be reduced to the case l = 0 by a change of parameters. Thus, Theorem 3.1 will follow from the following Lemma.
Lemma 3.2. Let n 1 , . . . , n r be non-negative integers, not all equal to zero, and let a 1 , . . . , a |n|−1 and c j (k), 1 ≤ j ≤ r, 0 ≤ k ≤ n j , be arbitrary scalars. Then n 1 ,...,nr Proof. The case |n| = 1 is easily verified. We thus assume |n| > 1, and prove the result by induction on |n|. Let f be the left-hand side of the identity, considered as a function of a 1 . We observe that f is a theta function of degree r + 1, nome p and norm 1. Moreover, if a 1 = c l (t), for some 1 ≤ l ≤ r and 0 ≤ t ≤ n l , the term with k l = t vanishes. The sum is then reduced to a sum of the same form, but with n l replaced by n l − 1, and with a 1 and c l (t) deleted from the parameter sequences. By the induction hypothesis, that sum vanishes. This shows that f has r + |n| zeroes, which may without loss of generality be considered generic. It follows from Lemma 2.2 that f is identically zero. Remark 3.3. In the proof we have implicitly assumed p = 0. When p = 0, the argument involving theta functions can be replaced by a polynomial argument. Namely, in that case f is a polynomial of degree r + 1 with r + |n| > r + 1 different zeroes, and thus vanishes identically. The resulting proof is different from the one given in [45].
Next we give a matrix inversion that we associate with the root system BC r . It contains a matrix inversion from [5] as a special case, see §3.1.
Theorem 3.4 (An elliptic BC r matrix inversion). Let (a t ) t∈Z and (c j (k)) k∈Z , 1 ≤ j ≤ r, be arbitrary sequences of scalars. Then the lower-triangular matrices Theorem 3.4 can be obtained as a special case of [45,Thm. 3.1], that is, of the trigonometric case of Theorem 3.1. More precisely, let p = 0 in Theorem 3.1, replace a t by ba t and c j (k) by bc j (k) and then let b → 0. This gives the matrix inversion .
Let us now reintroduce the elliptic nome by substituting After applying (2.3) to all factors, the auxiliary variables u and v only appear in factors that can be cancelled from (3.1). This eventually leads to Theorem 3.4. It is also possible to prove Theorem 3.4 directly in a similar way as Theorem 3.1. The analogue of Lemma 3.2 is then the identity n 1 ,...,nr The special case when n j ≡ 1 and c j (0)c j (1) is independent of j is equivalent to [36,Lemma 7.8]. Our proof of Theorem 3.1 is a straight-forward applications of Rains' method for proving that special case. Finally, we give a matrix inversion that we associate with the root system C r . The case p = 0 of Theorem 3.5 is [45, Thm. 4.1].
Theorem 3.5 (An elliptic C r matrix inversion). Let (c j (t)) t∈Z , 1 ≤ j ≤ r and b be arbitrary scalars. Then the lower-triangular matrices Note that the one-dimensional case of Theorem 3.5 is less general than for Theorems 3.1 and 3.4. Namely, it corresponds to the special case when the sequences a t and c 1 (t) are proportional. We have not been able to extend Theorem 3.5 to the level of generality of the other two inversions. Unfortunately, this means that our method of proof of those results does not apply to Theorem 3.5. Instead we give another proof which is more tedious in its details. The same method can be used to give alternative proofs of Theorem 3.1 and Theorem 3.4.
As before, to prove Theorem 3.5 it is enough to verify (3.1) for n > l = 0, which we state as a Lemma.
Lemma 3.6. Let n 1 , . . . , n r be non-negative integers, not all equal to zero, and let b and c j (k), 1 ≤ j ≤ r, 0 ≤ k ≤ n j , be arbitrary scalars. Then n 1 ,...,nr Proof. Pulling out all factors independent of k r , our sum can be written as .
We now rewrite the inner sum in (3.2) using Gustafson's identity (2.4). and λ = b/c 1 (k 1 ) · · · c r−1 (k r−1 ). Then (3.3) splits naturally into r parts, r i=1 S i = 0, where S 1 is the sum of the first n 1 terms, S 2 the following n 2 terms and so on until the final sum S r which has n r + 1 terms. It is easy to check that S r equals the inner sum in (3.2), so that We will complete the proof by showing that n 1 ,...,n r−1 By a symmetry argument, it suffices to do this for i = 1. We have After some elementary manipulations, we obtain where γ = b/c 2 (k 2 ) · · · c r (k r ). This has been written so as to emphasize the symmetry k 1 ↔ l. Indeed, all factors are symmetric under this change of variables, except for c 1 (k 1 )θ(c 1 (l)/c 1 (k 1 )), which is anti-symmetric. Thus, which completes the proof of the theorem.

Specializations of the matrix inversions.
For applications to hypergeometric series, one typically specializes the parameters in the matrix inversions to geometric progressions. For ease of reference, we give some specializations of this kind explicitly. We first consider the special case of Theorem 3.1 when a t = aq t and c j (k) = x j q mk (1 ≤ j ≤ r), with m a non-zero integer. It will be convenient to let This is another pair of lower-triangular inverse matrices, which has been normalized so that F n0 = G n0 = 1 for n ≥ 0. Let us first assume that m is positive. After some elementary manipulations, the corresponding special case of Theorem 3.1 takes the following form.
Corollary 3.7. When m is a positive integer, the lower-triangular matrices The case when m is negative can be obtained from Corollary 3.7 simply using the natural interpretation (a) −n = (aq −n ) −1 n for elliptic shifted factorials with negative subscripts. If we replace m by −m and x j by x −1 j for j = 1, . . . , r, the resulting identity takes the following form.
Corollary 3.8. When m is a positive integer, the lower-triangular matrices where X = x 1 · · · x r , are mutually inverse.
Finally, we let a t = aq t and c j (k) = x j q mk (1 ≤ j ≤ r) in Theorem 3.4. Since this result is symmetric under simultaneous inversion of all the parameters c j (k), we may assume that m is positive. This gives the following result.
Corollary 3.9. When m is a positive integer, the lower-triangular matrices are mutually inverse.
Since Theorem 3.5 lacks the parameters corresponding to a t in the other two inversions, we cannot give a specialization at the same level of generality. If we let c j (k) = x j q k (1 ≤ j ≤ r) we obtain an inversion equivalent to the case m = 1 of Corollary 3.9, but with F and G interchanged.
One can check that the explicit matrix inversions obtained by Bhatnagar and the second author [5, Cor. 4.5, Cor. 5.8, Cor. 9.6] are equivalent to the case m = 1 of Corollary 3.7, Corollary 3.8 and Corollary 3.9, respectively.

4.1.
Overview. If F and G are mutually inverse lower-triangular matrices, and there is a known summation formula 0≤k≤n F nk a k = b n , (4.1) then one can immediately deduce the inverse summation 0≤k≤n G nk b k = a n . (4.2) We will obtain several new results by applying this procedure to the multiple Jackson summations (2.5) and (2.6).
In the case of the A r Jackson summation (2.5) this can be done in four ways. Namely, it can be identified (in general, or in some special case) with (4.1), where F nk is as in Corollary 3.7 or as in Corollary 3.8, in both cases with either m = 1 or m = 2. In the case m = 1 of Corollary 3.8, the identities (4.1) and (4.2) are equivalent. In the other three cases (4.2) gives new summations, see §4.2.
The C r Jackson summation (2.6) can be expressed as (4.1), where F nk is as in Corollary 3.9, with 1 ≤ m ≤ 4. The sums (4.2) are then of a form traditionally associated with the root system D r (the motivation for this terminology is weak, but we find it convenient and will use it). The case m = 1 gives a new proof of the elliptic D r Jackson summation due to the first author [39,Cor. 6.3] (which for p = 0 is due independently to Bhatnagar [3] and the second author [45]). This is parallel to the proof in [45], and we do not provide the details. The remaining three cases lead to new summations, see §4.3.

New
In the inverse identity (4.2), we make the change of variables x i → tx i , i = 1, . . . , r, a → bcdt/aq, b → aq/bc and d → aq/bdt, where t is chosen so that after these substitutions t r+1 = a 2 q/bcdX. After simplification, we can eliminate t and arrive at the following result.
Theorem 4.1 (A new A r Jackson summation). We have the summation formula we have the summation formula where C = c 1 · · · c r+1 and X = x 1 · · · x r .
We sketch the standard argument for deducing Corollary 4.2 from Theorem 4.1. One first observes that the case c j = q −n j /x j (for 1 ≤ j ≤ r) of Corollary 4.2 is equivalent to the case c = q −N of Theorem 4.1. One then uses the quasi-periodicity (2.2) to deduce that Corollary 4.2 holds whenever c j x j ∈ p Z q Z ≤0 . The general case then follows by analytic continuation in the parameters c j .
The case p = 0 of Theorem 4.1 and Corollary 4.2 are due to the second author [49,Thm. 4.1,Cor. 4.2]. They generalize multiple 6 φ 5 summations due to Bhatnagar [2]. The general case of Theorem 4.1 has already been announced and applied in several publications. In [40], the first author found that it appears in connection with Felder's SU(2) elliptic quantum group. This led to an explicit system of multivariable biorthogonal functions, which essentially have the summand in (4.3) as their weight function. In [5, Thm. 5.2], Bhatnagar and the second author derived a multiple Bailey transformation 1 by combining (2.5) and (4.3) (the same result was independently obtained by the first author). Finally, in [41,Thm. 4.1] the first author found another multiple Bailey transformation by combining (4.3) with yet another (Gustafson-Rakha-type) multiple Jackson summation. In fact, we have found a third multiple Bailey transformation by combining (4.3) with itself, but we save that result for future work.
Next, we replace q by q 2 in (2.5) and then make the substitutions (a, b, c) → (X, aXq |n| , aXq |n|+1 ). Using that, in general, (a, aq; q 2 ) k = (a; q) 2k , we find that (4.1) holds, where F nk is as in the case m = 2 of Corollary 3.7 and where e 2 (k) = 1≤i<j≤r k i k j . In the inverse identity (4.2), we make the change of variables x i → tx i , i = 1, . . . , r, a → dt/aq and d → abq/dt, where t is chosen so that after these substitutions t r+1 = a 2 q/dX. After simplification, this gives the following result. Theorem 4.3. If a 2 q 2|n|+1 = cd, we have the quadratic summation formula By a standard argument (see the discussion of Corollary 4.2 above) we deduce the following companion identity.
Finally, we replace q by q 2 in (2.5) and then make the substitutions (a, d, e) → (X, aXq |n| , aXq |n|+1 ). The resulting identity takes the form (4.1), where F nk is as in the case m = 2 of Corollary 3.8 and This holds when a 2 bc = X 2 q. In the inverse identity (4.2), we make the substitutions t is chosen so that after these substitutions t r+1 = qa 2 /cX. This gives the following quadratic summation formula.
Corollary 4.6. If a 2 q = b 1 · · · b r+2 x 1 · · · x r , then The p = 0 cases of Theorem 4.5 and Corollary 4.6 are due to the second author [45,Thm. 5.4,Thm. 5.8]. In the case r = 1, they are equivalent to Theorem 4.3 and hence again reduce to summations from [53].

D r summations.
A D r quadratic summation formula is obtained if we replace q by q 2 in (2.6) and then make the substitutions (a, b, c, d, e) → (1, q/ab, b/a, aq |n| , aq |n|+1 ). The resulting identity takes the form (4.1), where F nk is as in the case m = 2 of Corollary 3.9 and The inverse identity (4.2) is easily simplified as follows.
Theorem 4.7. We have the quadratic summation formula (ax i q; q) 2n i (x i q/ab, bx i /a; q 2 ) n i (x i /a; q) 2n i (abx i q, ax i q 2 /b; q 2 ) n i . Theorem 4.7 has two companion identities where the summation is on a simplex. In the first one, we assume that b = q −N and use the standard argument to replace the parameters q −2n j /x j by generic parameters b j .
Corollary 4.8. We have the quadratic summation formula In the second companion identity to Theorem 4.7, we first let a = q −N − 1 2 and then replace x i by aq N + 1 2 x i for 1 ≤ i ≤ r, before applying the standard argument.
The cases p = 0 of Theorem 4.7, Corollary 4.8 and Corollary 4.9 are due to the second author [45,Thm. 5.21,Thm. 5.25,Thm 5.27]. In the case r = 1, they are equivalent to Theorem 4.3 and hence again reduce to summations from [53].
A D r cubic summation formula is obtained if we replace q by q 3 in (2.6) and then make the substitutions (a, b, c, d, e) → (1, aq |n| , aq |n|+1 , aq |n|+2 , 1/a 3 ). The resulting identity takes the form (4.1), where F nk is as in the case m = 3 of Corollary 3.9 and The inverse identity (4.2) is easily simplified as follows.
Theorem 4.10. We have the cubic summation formula We will give two companion identities to Theorem 4.10. For the first one, we first let a = q N/2 and then make the substitutions x i → ax i q −N/2 for 1 ≤ i ≤ r. The standard argument gives the following identity.
Corollary 4.11. We have the cubic summation formula Next, we let a = q −(N +1)/2 in Theorem 4.10, so that the factor (a 2 q; q) 2|k| vanishes unless 2|k| ≤ N. As before, this gives the following companion identity.
Corollary 4.12. We have the cubic summation formula The cases p = 0 of Theorem 4.10, Corollary 4.11 and Corollary 4.12 are due to the second author [45,Thm. 5.29,Thm. 5.34]. The case r = 1 of Theorem 4.10 is equivalent to the case b = a of [53,Cor. 4.5] and the corollaries to [53,Cor. 4.14 and Cor. 4.12], respectively. The case when both p = 0 and r = 1 of all these results are due to Gasper [18].