Complex binomial theorem and pentagon identities

N. M. Belousov^ab, G. A. Sarkissian^cd, V. P. Spiridonov<sup>ac

a</sup> International laboratory for Mirror Symmetry and Automorphic Forms, National Research University "Higher School of Economics", Moscow, Russia
^b Beijing Institute of Mathematical Sciences and Applications, Beijing, China
^c Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia
^d Yerevan Physics Institute, Yerevan, Armenia

Abstract: 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 $\omega_1+\omega_2\to 0$ (or $b\to i$ in two-dimensional conformal field theory) applied to hyperbolic hypergeometric integrals.

Keywords: binomial theorem, beta integrals, pentagonal identities, complex hypergeometric functions.

Received: 10.06.2025
Revised: 10.06.2025

DOI: 10.4213/tmf11022

 English version: Theoretical and Mathematical Physics, 2026, 226:1, 1–20

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