Abstract
The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.
Highlights
The Chebyshev polynomial Tn ( x ) see Section (22:3) in [1] has an algebraic definition given by: n n i p p 1 h Tn ( x ) =x + i 1 − x2 + x − i 1 − x2Attribution (CC BY) license.the polynomial has a Gauss hypergeometric function representation given by n Tn (1 − 2x ) = (−n) j (n) j∑ (1/2) j (1) j x j, j =0 the polynomial can be expressed by the Rodrigue’s formula
In this work we propose a systematic approach to deriving a quadruple integral involving a special function and derived it in terms of the Hurwitz-Lerch zeta function
We investigate the quadruple integral of the Chebyshev polynomial Tn ( x )
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In this work we propose a systematic approach to deriving a quadruple integral involving a special function and derived it in terms of the Hurwitz-Lerch zeta function. This approach is based on the contour integral method in [7]. The goal is to expand upon work on multiple integrals involving special functions with the aim of assisting researchers where this work is useful. In this present work, we investigate the quadruple integral of the Chebyshev polynomial Tn ( x ). Derive an integral transform which is invariant under the index n with respect to the Hurwitz-Lerch zeta function
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