Complex binomial theorem and pentagon identities

TL;DR

This paper explores pentagon identities via hyperbolic hypergeometric functions, deriving a complex binomial theorem and Fourier transform formulas for complex gamma functions through novel degenerations and limits.

Contribution

It introduces new degenerations of pentagon identities to complex hypergeometric functions and derives a complex binomial theorem and Fourier transform formulas for the complex gamma function.

Findings

Derived a complex binomial theorem from pentagon identities.

Established a Fourier transform formula for the complex gamma function.

Introduced a new limit approach in hypergeometric integrals.

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