Elliptic hypergeometric series

In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn1 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.

For surveys of elliptic hypergeometric series see Gasper & Rahman (2004), Spiridonov (2008) or Rosengren (2016).


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

Further reading

  • Spiridonov, V. P. (2013). "Aspects of elliptic hypergeometric functions". In Berndt, Bruce C. (ed.). The Legacy of Srinivasa Ramanujan Proceedings of an International Conference in Celebration of the 125th Anniversary of Ramanujan's Birth ; University of Delhi, 17-22 December 2012. Ramanujan Mathematical Society Lecture Notes Series. 20. Ramanujan Mathematical Society. pp. 347–361. arXiv:1307.2876. Bibcode:2013arXiv1307.2876S. ISBN 9789380416137.
  • Rosengren, Hjalmar (2016). "Elliptic Hypergeometric Functions". arXiv:1608.06161 [math.CA].


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