Complex binomial theorem and pentagon identities

TL;DR

The work develops a systematic degeneration of hyperbolic pentagon identities to complex hypergeometric structures, revealing a complex binomial theorem as a $b\to\text{i}$ limit of a reduced pentagon identity and connecting it to the complex analogue of Euler’s beta integral via Fourier-type transforms. A new degeneration limit for ratios of hyperbolic gamma functions is introduced to bridge hyperbolic and complex levels, yielding a Mellin–Barnes form for the complex beta integral and establishing a network of identities among complex gamma functions, pentagon relations, and star-triangle relations. The results unify several strands from hyperbolic and elliptic hypergeometric theories, two-dimensional conformal field theory, and topological field theories, and provide tools for complex integrable systems such as Ruijsenaars-type models. These complex degenerations illuminate deep structural links between pentagon identities, Fourier transforms, and beta-type integrals, with potential applications in mathematical physics and quantum topology.

Abstract

We consider different pentagon identities realized by the 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 analogue of the Euler beta integral. 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 $ω_1+ω_2\to 0$ (or $b\to \textrm{i}$ in two-dimensional conformal field theory) applied to the hyperbolic hypergeometric integrals.