In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They were introduced by Date-Jimbo-Kuniba-Miwa-Okado (1987) and in their study of elliptic 6-j symbols. For surveys of elliptic hypergeometric series see, or.

Definitions

The q-Pochhammer symbol is defined by The modified Jacobi theta function with argument x and nome p is defined by The elliptic shifted factorial is defined by The theta hypergeometric series r+1Er is defined by The very well poised theta hypergeometric series r+1Vr is defined by The bilateral theta hypergeometric series rGr is defined by

Definitions of additive elliptic hypergeometric series

The elliptic numbers are defined by where the Jacobi theta function is defined by The additive elliptic shifted factorials are defined by The additive theta hypergeometric series r+1er is defined by The additive very well poised theta hypergeometric series r+1vr is defined by