Multi-parameter Mathieu, and alternating Mathieu series. (English) Zbl 1508.33007
Summary: The main purpose of this paper is to present a multi-parameter study of the familiar Mathieu series and the alternating Mathieu series \(\mathcal{S}(\boldsymbol{r})\) and \(\widetilde{\mathcal{S}}(\boldsymbol{r})\). The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella’s hypergeometric function \(F_C^{(m)}[\boldsymbol{x}]\), 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 Turán 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 multi-parameter treatment of the complete Butzer-Flocke-Hauss \(\Omega\) function which is intimately connected with the Mathieu series family.
MSC:
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26D15 | Inequalities for sums, series and integrals |
| 33C65 | Appell, Horn and Lauricella functions |
| 33E05 | Elliptic functions and integrals |
| 33E20 | Other functions defined by series and integrals |
| 44A10 | Laplace transform |
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; Turán inequalityReferences:
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