Elliptic hypergeometric series

(Redirected from Theta 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

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

 

Further reading

edit
  • 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

edit