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 In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...

of ''n'', analogous to generalized hypergeometric series where the ratio is a

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...

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 symbol Wigner's 6-''j'' symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols, : \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end = \sum_ (-1)^ \beg ...

s. For surveys of elliptic hypergeometric series see , or .

Definitions

The

q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

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 :

\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,\tau n=;\sigma,\tau a+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

References

* * * * * * {{series (mathematics) Hypergeometric functions

Definitions

Definitions of additive elliptic hypergeometric series

Further reading

References