New identities obtained from Gegenbauer series expansion

TL;DR

This paper derives new mathematical identities using Fourier-Gegenbauer series expansions, extending known results and providing a novel formula involving tangent functions and binomial coefficients for complex numbers.

Contribution

It introduces new identities from Gegenbauer series expansions, generalizing existing results and connecting series with tangent functions for complex variables.

Findings

Proved a new identity involving binomial coefficients and tangent functions.

Extended known series identities to complex domains.

Established conditions for the validity of the derived identities.

Abstract

Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*} \sum_{n=0}^\infty\frac{1}{4^n}\binom{2n}{n}\frac{z-2n}{\binom{z-1/2}{n}}\binom{z}{n}^3 =\frac{\tan(\pi z)}{\pi} \end{equation*} for all complex numbers $z$ such that $\Re(z)>-\frac{1}{2}$ and $z\notin\frac{1}{2}+\mathbb{Z}$.