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Multi-parameter Mathieu, and alternating Mathieu series
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DOI：10.1016/j.amc.2021.126099.png)
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Abstract
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The main purpose of this paper is to present a multi-parameter study of the familiar Mathieu series and the alternating Mathieu series F (r) and ( F) over tilde (r). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella's hypergeometric function F-C((m)) [x], Fox-Wright Psi function, Srivastava-Daoust S generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper-Bessel and Bessel-Clifford functions. Auxiliary Laplace-Mellin transforms, bounding inequalities for the hyper-Bessel and Bessel-Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper-Bessel and Bessel-Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu-type series. Next, functional bounding inequalities, log-convexity properties and Turan inequality results are presented for the investigated extensions of multi-parameter Mathieu-type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multiparameter treatment of the complete Butzer-Flocke-Hauss Omega function which is intimately connected with the Mathieu series family. (C) 2021 Elsevier Inc. All rights reserved.
Keywords:
Mathieu and alternating Mathieu series
Lauricellas hypergeometric functions
Generalized Weber-Schafheitlin integral
Riemann Zeta function
Dirichlet Eta function
Laplace transforms
Mellin transforms
Functional bounding inequalities
log-convexity
Turan inequality
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