formulae expressing explicitly the coefficients of an expansion of double jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. extension to expansion of triple jacobi polynomials is given. the results for the special cases of double and triple ultraspher...

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...

Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, t...

We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hy-pergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

Elliptic hypergeometric series form a natural generalization of hypergeometric and basic hypergeometric (or q-) series. It is surprising that they were introduced only very recently, by Frenkel and Turaev [FT], who expressed the 6j-symbols corresponding to certain elliptic solutions of the Yang–Baxter equation, cf. [DJ], in terms of the 10ω9-sums defined below. It is expected that elliptic hype...

We prove a master theorem for hypergeometric functions of Karlsson–Minton type, stating that a very general multilateral U(n) Karlsson–Minton type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson–Minton summation formula and many of its known generalizations as special cases, and it also implies several “Bailey-type” identities for U(n) hypergeometric se...