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Abstract <p> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega_1+\omega_2\to 0\)</EquationSource> </InlineEquation> (or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(b\to i\)</EquationSource> </InlineEquation> in two-dimensional conformal field theory) applied to hyperbolic hypergeometric integrals. </p>

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Complex binomial theorem and pentagon identities

  • 期刊论文

  • 发表日期：

    2026年1月26日

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  • N. M. Belousov,
  • G. A. Sarkissian,
  • V. P. Spiridonov

摘要

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.