Gustafson-Rakha-Type Elliptic Hypergeometric Series

Abstract

Abstract. 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.

Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

“…After discovering this result, the authors were informed by Rosengren [33] that he obtained the same result by following the approach used in [10,31]. Other elliptic Bailey transformations on root systems were given previously by Rosengren [31,34].…”

“…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.) One of the goals of our study is to recover the extensions of Bailey's transformation formula listed by Rosengren [31], which were obtained by a straightforward extension of the approach followed in [10] for the basic hypergeometric case.…”

Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems

“…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.…”

Hypergeometric and basic hypergeometric series and integrals associated with root systems

“…The following A n Jackson summation is due to Gustafson and Rakha [41, Theorem 1.2] (but stated here as in [110] where it has been extended to the elliptic level):…”