Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems

Abstract

Multidimensional matrix inversions provide 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.

Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

“…Rosengren and the second author [35] have proved more general elliptic matrix inversions, which contain the inverses of most of the matrices in this paper.…”

“…One of our Bailey lemmas is equivalent to a result of Zhang and Huang [51]. Most of the matrix inversions that appear in our work can be obtained as special cases of very general matrix inversions due to Rosengren and the second author [35]. Warnaar [48] found four of the WP Bailey lemmas (including the one due to Zhang and Huang [51]) a few years ago but did not publish his work.…”

“…Our work in this paper depends on the elliptic A n , C n and D n generalizations of this summation due to Rosengren [31] and one such result due to Rosengren and the second author [35]. (Recently, Rosengren [34] gave yet another such result, which we do not include in our study.)…”

“…The A n elliptic Jackson summation we use in this section is due to Rosengren and the second author [35]. The p " 0 case is due to the second author [38].…”

An Elliptic Hypergeometric Function Approach to Branching Rules

“…Two of these play a key role in our proof of Theorem 1.2. We remark that higher-dimensional analogues of a different type of quadratic elliptic hypergeometric series, in which the term (apq; q, p) 2k /(a; q, p) 2k in (2.17) is replaced by (apq; q, p) 3k /(a; q, p) 3k , were recently considered in [44].…”