Some families of Mathieu a-series and alternating Mathieu a-series

• Published in: Applied mathematics and computation Vol. 173; no. 1; pp. 69 - 108
• Main Authors: Pogány, Tibor K., Srivastava, H.M., Tomovski, Živorad
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.02.2006; Elsevier
• Subjects: Algebra; Asymptotic expansions; Bernoulli numbers; Bessel function of the first kind; Bounding inequalities; Dirichlet Eta function; Dirichlet series; Euler–Maclaurin summation formula; Exact sciences and technology; Fourier transforms; Fox–Wright Psi-functions; Fredholm integral equation; Generalized Mathieu series; Genocchi numbers; Hypergeometric functions; Integral equations; Integral representations; Integral transforms, operational calculus; Landau bounds for Jν( x); Laplace integrals; Mathematical analysis; Mathematics; Mathieu and alternating Mathieu a-series; Mathieu and alternating Mathieu series; Mellin transforms; Number theory; Riemann Zeta function; Sciences and techniques of general use; Special functions; Weber–Sonine integral; Hypergeometric functions; Landau bounds for J ν ( x); Asymptotic expansions; Fourier transforms; Bounding inequalities; Weber–Sonine integral; Mathieu and alternating Mathieu a-series; Bernoulli numbers; Fredholm integral equation; Bessel function of the first kind; Euler–Maclaurin summation formula; Fox–Wright Psi-functions; Mathieu and alternating Mathieu series; Generalized Mathieu series; Mellin transforms; Integral representations; Dirichlet series; Laplace integrals; Riemann Zeta function; Dirichlet Eta function; Genocchi numbers; Bessel function; Fourier transformation; Dirichlet eta function; Mathieu series; Weber-Sonine integral; Hypergeometric function; Integral equation; Fox-Wright Psi-functions; Mellin transformation; Convergence acceleration; Wright function; Special function; Fredholm equation; Numerical analysis; Landau bound; Asymptotic expansion; Riemann function; Applied mathematics; Euler-Maclaurin summation formula; Zeta function; Bernoulli number; Integral representation; Summation; Landau bounds for Jv(x)
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2005.02.044

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu a-series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu a-series. Relationships of the Mathieu a-series with the Riemann Zeta function and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function J ν ( z) and the confluent hypergeometric functions 0 F 1 and 1 F 2 are characterized by means of certain Fredholm type integral equations of the first kind, which are associated with some of these Mathieu type series. Several integrals containing Mathieu type series are also evaluated. Finally, some closely-related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.