# 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
    $\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}).$
    $\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.$
The modified Jacobi theta function with argument x and nome p is defined by
    $\displaystyle \theta (x;p)=(x,p/x;p)_{\infty }$
    $\displaystyle \theta (x_{1},...,x_{m};p)=\theta (x_{1};p)...\theta (x_{m};p)$
The elliptic shifted factorial is defined by
    $\displaystyle (a;q,p)_{n}=\theta (a;p)\theta (aq;p)...\theta (aq^{n-1};p)$
    $\displaystyle (a_{1},...,a_{m};q,p)_{n}=(a_{1};q,p)_{n}\cdots (a_{m};q,p)_{n}$
The theta hypergeometric series $_{r+1}$E$_r$ is defined by
    $\displaystyle_{r+1}E_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};q,p;z)=\sum _{n=0}^{\infty }{\frac {(a_{1},...,a_{r+1};q;p)_{n}}{(q,b_{1},...,b_{r};q,p)_{n}}}z^{n}$
The very well poised theta hypergeometric series $_{r+1}$V$_r$ is defined by
    $\displaystyle_{r+1}V_{r}(a_{1};a_{6},a_{7},...a_{r+1};q,p;z)=\sum _{n=0}^{\infty }{\frac {\theta (a_{1}q^{2n};p)}{\theta (a_{1};p)}}{\frac {(a_{1},a_{6},a_{7},...,a_{r+1};q;p)_{n}}{(q,a_{1}q/a_{6},a_{1}q/a_{7},...,a_{1}q/a_{r+1};q,p)_{n}}}(qz)^{n}$
The bilateral theta hypergeometric series $_r$G$_r$ is defined by
    $\displaystyle_{r}G_{r}(a_{1},...a_{r};b_{1},...,b_{r};q,p;z)=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},...,a_{r};q;p)_{n}}{(b_{1},...,b_{r};q,p)_{n}}}z^{n}$

## 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
    $\theta _{1}(x,q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(n+1/2)^{2}}e^{(2n+1)ix}$
The additive elliptic shifted factorials are defined by
* $[a;\sigma ,\tau ]_{n}=[a;\sigma ,\tau ][a+1;\sigma ,\tau ]...[a+n-1;\sigma ,\tau ]$
* $[a_{1},...,a_{m};\sigma ,\tau ]=[a_{1};\sigma ,\tau ]...[a_{m};\sigma ,\tau ]$
The additive theta hypergeometric series $_{r+1}$e$_r$ is defined by
    $\displaystyle_{r+1}e_{r}(a_{1},...a_{r+1};b_{1},...,b_{r};\sigma ,\tau ;z)=\sum _{n=0}^{\infty }{\frac {[a_{1},...,a_{r+1};\sigma ;\tau ]_{n}}{[1,b_{1},...,b_{r};\sigma ,\tau ]_{n}}}z^{n}$
The additive very well poised theta hypergeometric series $_{r+1}$v$_r$ is defined by
    $\displaystyle_{r+1}v_{r}(a_{1};a_{6},...a_{r+1};\sigma ,\tau ;z)=\sum _{n=0}^{\infty }{\frac {[a_{1}+2n;\sigma ,\tau ]}{[a_{1};\sigma ,\tau ]}}{\frac {[a_{1},a_{6},...,a_{r+1};\sigma ,\tau ]_{n}}{[1,1+a_{1}-a_{6},...,1+a_{1}-a_{r+1};\sigma ,\tau ]_{n}}}z^{n}$