Elliptic hypergeometric series

In mathematics, an elliptic hypergeometric series is a series Σc_n such that the ratio c_n/c_n−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).

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+1E_r is defined by

The very well poised theta hypergeometric series _r+1V_r is defined by

The bilateral theta hypergeometric series _rG_r is defined by

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+1e_r is defined by

The additive very well poised theta hypergeometric series _r+1v_r is defined by

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