Summation formulae for elliptic hypergeometric series

Summation Formulae for Noncommutative Hypergeometric Series

Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have recently been studied by a number of researchers including (in alphabetical order) Durán, Duval, Grünbaum, Iliev, Ovsienko, Pacharoni, Tirao, and others. See [3, 6, 8, 9, 10, 11, 12, 16] for some selected papers. The subject of hypergeometric series involving matrices is closel...

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

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.

Elliptic Hypergeometric Series on Root Systems

We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems An, Cn and Dn. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities.

Hypergeometric Series and Periods of Elliptic Curves

In [7], Greene introduced the notion of general hypergeometric series over finite fields or Gaussian hypergeometric series, which are analogous to classical hypergeometric series. The motivation for his work was to develop the area of character sums and their evaluations through parallels with the theory of hypergeometric functions. The basis for this parallel was the analogy between Gauss sums...