Abstract
A four-dimensional integral containing g(x,y,z,t)Cn(λ)(x) is derived. Cn(λ)(x) 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≤x≤1,0≤y≤1,0≤z≤1,0≤t≤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.
Highlights
The definite integral of the Gegenbauer polynomial is evaluated in the work by Askey et al [1]
The Gegenbauer polynomial has many mathematical applications which are detailed in Andrews et al [4] (1999, Chapter 9)
The method used by us in [6] are followed in the derivations. This method employs a form of the generalized Cauchy’s integral formula, which is given by creativecommons.org/licenses/by/
Summary
Quadruple Integral Containing the Gegenbauer Polynomial Cn ( x ): Derivation and Evaluation. The definite integral of the Gegenbauer polynomial is evaluated in the work by Askey et al [1]. In the work by Srivastava [2] the author obtained an inversion formula for a singular integral transform involving Gegenbauer polynomials. The Gegenbauer polynomial has many mathematical applications which are detailed in Andrews et al [4] (1999, Chapter 9). Other applications are detailed in section (18.39) in [5]. We extend the previous important work by adding three more dimensions to the previously derived integrals in this paper. A quadruple integral will be derived and expressed in terms of a Hurwitz–Lerch zeta function. Hurwitz–Lerch zeta function zeta(s, v), the digamma function psi (0)(s), the Riemann zeta function zeta(k), and log(2) are used to deduce special cases
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