Elliptic gamma function

In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by Jackson (1905), and can be expressed in terms of the triple gamma function. It is given by

Γ (z; p, q) = \prod_{m = 0}^{\infty} \prod_{n = 0}^{\infty} \frac{1 - p^{m + 1} q^{n + 1} / z}{1 - p^{m} q^{n} z} .

It obeys several identities:

Γ (z; p, q) = \frac{1}{Γ (p q / z; p, q)}

Γ (p z; p, q) = θ (z; q) Γ (z; p, q)

and

Γ (q z; p, q) = θ (z; p) Γ (z; p, q)

where θ is the q-theta function.

When $p = 0$ , it essentially reduces to the infinite q-Pochhammer symbol:

Γ (z; 0, q) = \frac{1}{(z; q)_{\infty}} .

Multiplication Formula

Define

\tilde{Γ} (z; p, q) := \frac{(q; q)_{\infty}}{(p; p)_{\infty}} (θ (q; p))^{1 - z} \prod_{m = 0}^{\infty} \prod_{n = 0}^{\infty} \frac{1 - p^{m + 1} q^{n + 1 - z}}{1 - p^{m} q^{n + z}} .

Then the following formula holds with $r = q^{n}$ (Felder & Varchenko (2002)).

\tilde{Γ} (n z; p, q) \tilde{Γ} (1 / n; p, r) \tilde{Γ} (2 / n; p, r) \dots \tilde{Γ} ((n - 1) / n; p, r) = {(\frac{θ (r; p)}{θ (q; p)})}^{n z - 1} \tilde{Γ} (z; p, r) \tilde{Γ} (z + 1 / n; p, r) \dots \tilde{Γ} (z + (n - 1) / n; p, r) .

Felder, G.; Varchenko, A. (2002). "Multiplication Formulas for the Elliptic Gamma Function". arXiv:math/0212155.
Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 76 (508), The Royal Society: 127–144, Bibcode:1905RSPSA..76..127J, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, JSTOR 92601
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
Ruijsenaars, S. N. M. (1997), "First order analytic difference equations and integrable quantum systems", Journal of Mathematical Physics, 38 (2): 1069–1146, Bibcode:1997JMP....38.1069R, doi:10.1063/1.531809, ISSN 0022-2488, MR 1434226
Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal. 141. arXiv:math/0601337. doi:10.1215/S0012-7094-08-14111-0. S2CID 817920.