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.

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

Definitions

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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

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The elliptic numbers are defined by

 

where the Jacobi theta function is defined by

 

The additive elliptic shifted factorials are defined by

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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

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  • 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. Vol. 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].

References

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