Elliptic hypergeometric series

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+1E**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+1V**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_1q^{2n};p)}{\theta(a_1;p)}\frac{(a_1,a_6,a_7,...,a{r+1};q,p)n}{(q,a_1q/a_6,a_1q/a_7,...,a_1q/a{r+1};q,p)n}(qz)^n The bilateral theta hypergeometric series rGr 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)^nq^{(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+1e**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+1v**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

References