In mathematics, an elliptic hypergeometric series is a series such that the ratio is an elliptic function of , analogous to generalized hypergeometric series where the ratio is a rational function of , and basic hypergeometric series where the ratio is a periodic function of the complex number . 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 and nome is defined by

The elliptic shifted factorial is defined by

The theta hypergeometric series is defined by

The very well poised theta hypergeometric series is defined by

The bilateral theta hypergeometric series 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

Further reading

References