Elliptic hypergeometric series

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

For surveys of elliptic hypergeometric series see Gasper & Rahman (2004) or Spiridonov (2008).

Definitions[edit]

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[edit]

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

References[edit]