Elliptic hypergeometric series

In mathematics, an elliptic hypergeometric series is a series Σc_n such that the ratio c_n/c_n−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

The q-Pochhammer symbol is defined by

$(a;q{)}_n=\prod _{k=0}^{n-1}(1-a{q}^k)=(1-a)(1-aq)(1-a{q}^2)\cdots (1-a{q}^{n-1}).$

$(a_1,a_2,\dots ,a_m;q{)}_n=(a_1;q{)}_n(a_2;q{)}_n\dots (a_m;q{)}_n.$

The modified Jacobi theta function with argument x and nome p is defined by

$\theta (x;p)=(x,p/x;p{)}_{\mathrm{\infty }}$

$\theta (x_1,\mathrm{...},x_m;p)=\theta (x_1;p)\mathrm{...}\theta (x_m;p)$

The elliptic shifted factorial is defined by

$(a;q,p{)}_n=\theta (a;p)\theta (aq;p)\mathrm{...}\theta (a{q}^{n-1};p)$

$(a_1,\mathrm{...},a_m;q,p{)}_n=(a_1;q,p{)}_n\cdots (a_m;q,p{)}_n$

The theta hypergeometric series _r+1E_r is defined by

${}_{r+1}{E}_r(a_1,\mathrm{...}{a}_{r+1};b_1,\mathrm{...},b_r;q,p;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a_1,\mathrm{...},a_{r+1};q,p{)}_n}{(q,b_1,\mathrm{...},b_r;q,p{)}_n}{z}^n$

The very well poised theta hypergeometric series _r+1V_r is defined by

${}_{r+1}{V}_r(a_1;a_6,a_7,\mathrm{...}{a}_{r+1};q,p;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{\theta (a_1{q}^{2n};p)}{\theta (a_1;p)}\frac{(a_1,a_6,a_7,\mathrm{...},a_{r+1};q,p{)}_n}{(q,a_{1}q/a_6,a_{1}q/a_7,\mathrm{...},a_{1}q/a_{r+1};q,p{)}_n}(qz{)}^n$

The bilateral theta hypergeometric series _rG_r is defined by

${}_r{G}_r(a_1,\mathrm{...}{a}_r;b_1,\mathrm{...},b_r;q,p;z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(a_1,\mathrm{...},a_r;q,p{)}_n}{(b_1,\mathrm{...},b_r;q,p{)}_n}{z}^n$

Definitions of additive elliptic hypergeometric series

The elliptic numbers are defined by

$[a;\sigma ,\tau ]=\frac{{\theta }_1(\pi \sigma a,e^{\pi i\tau })}{{\theta }_1(\pi \sigma ,e^{\pi i\tau })}$

where the Jacobi theta function is defined by

${\theta }_1(x,q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{(n+1/2{)}^2}{e}^{(2n+1)ix}$

The additive elliptic shifted factorials are defined by

• $[a;\sigma ,\tau {]}_n=[a;\sigma ,\tau ][a+1;\sigma ,\tau ]\mathrm{...}[a+n-1;\sigma ,\tau ]$
• $[a_1,\mathrm{...},a_m;\sigma ,\tau ]=[a_1;\sigma ,\tau ]\mathrm{...}[a_m;\sigma ,\tau ]$

The additive theta hypergeometric series _r+1e_r is defined by

${}_{r+1}{e}_r(a_1,\mathrm{...}{a}_{r+1};b_1,\mathrm{...},b_r;\sigma ,\tau ;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{[a_1,\mathrm{...},a_{r+1};\sigma ,\tau {]}_n}{[1,b_1,\mathrm{...},b_r;\sigma ,\tau {]}_n}{z}^n$

The additive very well poised theta hypergeometric series _r+1v_r is defined by

${}_{r+1}{v}_r(a_1;a_6,\mathrm{...}{a}_{r+1};\sigma ,\tau ;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{[a_1+2n;\sigma ,\tau ]}{[a_1;\sigma ,\tau ]}\frac{[a_1,a_6,\mathrm{...},a_{r+1};\sigma ,\tau {]}_n}{[1,1+a_1-a_6,\mathrm{...},1+a_1-a_{r+1};\sigma ,\tau {]}_n}{z}^n$