A Quadruple Integral Containing the Gegenbauer Polynomial Cn(λ)(x): Derivation and Evaluation

Abstract

A four-dimensional integral containing $g(x,y,z,t)C_n^{\left(\lambda \right)}\left(x\right)$ is derived. $C_n^{\left(\lambda \right)}\left(x\right)$ is the Gegenbauer polynomial, $g(x,y,z,t)$ is a product of the generalized logarithm quotient functions and the integral is taken over the region $0\le x\le 1,0\le y\le 1,0\le z\le 1,0\le t\le 1$. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new.