Modular Hypergeometric Series

In mathematics, an elliptic hypergeometric series is a series Σ''c''_''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 in their study of elliptic 6-j symbols.

For surveys of elliptic hypergeometric series see , or .

Definitions

The q-Pochhammer symbol is defined by
:$\displaystyle(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^).$
:$\displaystyle(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n.$
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
:$\displaystyle(a;q,p)_n = \theta(a;p)\theta(aq;p)...\theta(aq^;p)$
:$\displaystyle(a_1,...,a_m;q,p)_n=(a_1;q,p)_n\cdots(a_m;q,p)_n$
The theta hypergeometric series _''r''+1''E''_''r'' is defined by
:$\displaystyle_E_r(a_1,...a_;b_1,...,b_r;q,p;z) = \sum_^\infty\fracz^n$
The very well poised theta hypergeometric series _''r''+1''V''_''r'' is defined by
:$\displaystyle_V_r(a_1;a_6,a_7,...a_;q,p;z) = \sum_^\infty\frac\frac(qz)^n$
The bilateral theta hypergeometric series _''r''''G''_''r'' is defined by
:$\displaystyle_G_r(a_1,...a_;b_1,...,b_r;q,p;z) = \sum_^\infty\fracz^n$

Definitions of additive elliptic hypergeometric series

The elliptic numbers are defined by
: ;\sigma,\tau\frac
where the Jacobi theta function is defined by
:$\theta_1(x,q) = \sum_^\infty (-1)^nq^e^$
The additive elliptic shifted factorials are defined by
* ;\sigma,\taun= ;\sigma,\taua+1;\sigma,\tau]... +n-1;\sigma,\tau/math>
* _1,...,a_m;\sigma,\tau= _1;\sigma,\tau.. _m;\sigma,\tau/math>
The additive theta hypergeometric series _''r''+1''e''_''r'' is defined by
:$\displaystyle_e_r(a_1,...a_;b_1,...,b_r;\sigma,\tau;z) = \sum_^\infty\fracz^n$
The additive very well poised theta hypergeometric series _''r''+1''v''_''r'' is defined by
:$\displaystyle_v_r(a_1;a_6,...a_;\sigma,\tau;z) = \sum_^\infty\frac\fracz^n$

Further reading

*
*

References

*
*
*
*
*
*

{{series (mathematics)

Hypergeometric functions