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 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
( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 1 ) . ( a 1 , a 2 , … , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n … ( a m ; q ) n . The modified Jacobi theta function with argument x and nome p is defined by
θ ( x ; p ) = ( x , p / x ; p ) ∞ θ ( x 1 , . . . , x m ; p ) = θ ( x 1 ; p ) . . . θ ( x m ; p ) The elliptic shifted factorial is defined by
( a ; q , p ) n = θ ( a ; p ) θ ( a q ; p ) . . . θ ( a q n − 1 ; p ) ( a 1 , . . . , a m ; q , p ) n = ( a 1 ; q , p ) n ⋯ ( a m ; q , p ) n The theta hypergeometric series r+1Er is defined by
r + 1 E r ( a 1 , . . . a r + 1 ; b 1 , . . . , b r ; q , p ; z ) = ∑ n = 0 ∞ ( a 1 , . . . , a r + 1 ; q ; p ) n ( q , b 1 , . . . , b r ; q , p ) n z n The very well poised theta hypergeometric series r+1Vr is defined by
r + 1 V r ( a 1 ; a 6 , a 7 , . . . a r + 1 ; q , p ; z ) = ∑ n = 0 ∞ θ ( a 1 q 2 n ; p ) θ ( a 1 ; p ) ( a 1 , a 6 , a 7 , . . . , a r + 1 ; q ; p ) n ( q , a 1 q / a 6 , a 1 q / a 7 , . . . , a 1 q / a r + 1 ; q , p ) n ( q z ) n The bilateral theta hypergeometric series rGr is defined by
r G r ( a 1 , . . . a r ; b 1 , . . . , b r ; q , p ; z ) = ∑ n = − ∞ ∞ ( a 1 , . . . , a r ; q ; p ) n ( b 1 , . . . , b r ; q , p ) n z n The elliptic numbers are defined by
[ a ; σ , τ ] = θ 1 ( π σ a , e π i τ ) θ 1 ( π σ , e π i τ ) where the Jacobi theta function is defined by
θ 1 ( x , q ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( n + 1 / 2 ) 2 e ( 2 n + 1 ) i x The additive elliptic shifted factorials are defined by
[ a ; σ , τ ] n = [ a ; σ , τ ] [ a + 1 ; σ , τ ] . . . [ a + n − 1 ; σ , τ ] [ a 1 , . . . , a m ; σ , τ ] = [ a 1 ; σ , τ ] . . . [ a m ; σ , τ ] The additive theta hypergeometric series r+1er is defined by
r + 1 e r ( a 1 , . . . a r + 1 ; b 1 , . . . , b r ; σ , τ ; z ) = ∑ n = 0 ∞ [ a 1 , . . . , a r + 1 ; σ ; τ ] n [ 1 , b 1 , . . . , b r ; σ , τ ] n z n The additive very well poised theta hypergeometric series r+1vr is defined by
r + 1 v r ( a 1 ; a 6 , . . . a r + 1 ; σ , τ ; z ) = ∑ n = 0 ∞ [ a 1 + 2 n ; σ , τ ] [ a 1 ; σ , τ ] [ a 1 , a 6 , . . . , a r + 1 ; σ , τ ] n [ 1 , 1 + a 1 − a 6 , . . . , 1 + a 1 − a r + 1 ; σ , τ ] n z n 