Complex binomial theorem and pentagon identities

N. M. Belousov

doi:10.1134/S0040577926010010

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Complex binomial theorem and pentagon identities

2026-01-01

PRE

N. M. Belousov

Gor Sarkissian

V. P. Spiridonov *

DOI：10.1134/S0040577926010010

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Abstract

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We consider various pentagon identities realized by hyperbolic hypergeometric functions and investigate their degenerations to the level of complex hypergeometric functions. In particular, we show that one of the degenerations yields the complex binomial theorem, which coincides with the Fourier transformation of the complex Euler beta integral evaluation. At the bottom, we obtain a Fourier transformation formula for the complex gamma function. This is done with the help of a new type of the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega_1+\omega_2\to 0$$\end{document} (or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\to i$$\end{document} in two-dimensional conformal field theory) applied to hyperbolic hypergeometric integrals.

Keywords:

binomial theorem

beta integrals

pentagonal identities

complex hypergeometric functions