Elliptic hypergeometric series

Template:Short description 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 Template:Harvtxt in their study of elliptic 6-j symbols.

For surveys of elliptic hypergeometric series see Template:Harvtxt, Template:Harvtxt or Template:Harvtxt.

Definitions

(a; q)_{n} = \prod_{k = 0}^{n - 1} (1 - a q^{k}) = (1 - a) (1 - a q) (1 - a q^{2}) \dots (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

θ (x; p) = (x, p / x; p)_{\infty}

θ (x_{1}, . . ., x_{m}; p) = θ (x_{1}; p) . . . θ (x_{m}; p)

The elliptic shifted factorial is defined by

(a; q, p)_{n} = θ (a; p) θ (a q; p) . . . θ (a q^{n - 1}; p)

(a_{1}, . . ., a_{m}; q, p)_{n} = (a_{1}; q, p)_{n} \dots (a_{m}; q, p)_{n}

The theta hypergeometric series _r+1E_r is defined by

_{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

_{r + 1} V_{r} (a_{1}; a_{6}, a_{7}, . . . a_{r + 1}; q, p; z) = \sum_{n = 0}^{\infty} \frac{θ (a_{1} q^{2 n}; p)}{θ (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}} (q z)^{n}

The bilateral theta hypergeometric series _rG_r is defined by

_{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}

The elliptic numbers are defined by

[a; σ, τ] = \frac{θ_{1} (π σ a, e^{π i τ})}{θ_{1} (π σ, e^{π i τ})}

where the Jacobi theta function is defined by

θ_{1} (x, q) = \sum_{n = - \infty}^{\infty} (- 1)^{n} q^{(n + 1 / 2)^{2}} e^{(2 n + 1) i x}

The additive elliptic shifted factorials are defined by

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

_{r + 1} e_{r} (a_{1}, . . . a_{r + 1}; b_{1}, . . ., b_{r}; σ, τ; z) = \sum_{n = 0}^{\infty} \frac{[a_{1}, . . ., a_{r + 1}; σ; τ]_{n}}{[1, b_{1}, . . ., b_{r}; σ, τ]_{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}, . . . a_{r + 1}; σ, τ; z) = \sum_{n = 0}^{\infty} \frac{[a_{1} + 2 n; σ, τ]}{[a_{1}; σ, τ]} \frac{[a_{1}, a_{6}, . . ., a_{r + 1}; σ, τ]_{n}}{[1, 1 + a_{1} - a_{6}, . . ., 1 + a_{1} - a_{r + 1}; σ, τ]_{n}} z^{n}