Elliptic hypergeometric series
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 Frenkel & Turaev (1997) in their study of elliptic 6-j symbols.
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
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