Integral representations and integral transforms of some families of Mathieu type series

Abstract

By using some integral representations for several Mathieu type series (see P.L. Butzer, T.K. Pogány, and H.M. Srivastava, A linear ODE for the Omega function associated with the Euler function E _α(z) and the Bernoulli function B _α(z), Appl. Math. Lett. 19 (2006), pp. 1073–1077; P. Cerone and C.T. Lenard, On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003), Article 100, pp. 1–11 (electronic), T.K. Pogány; H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a-series and alternating Mathieu a-series, Appl. Math. Comput. 173 (2006), pp. 69–108; H.M. Srivastava and Ž. Tomovski, Some problems and solutions involving Mathieu's series and its generalizations, J. Inequal. Pure Appl. Math. 5 (2) (2004), Article 45, pp. 1–13 (electronic); Ž. Tomovski, Integral representations of generalized Mathieu series via Mittag-Leffler type functions, Fract. Calc. Appl. Anal. 10 (2007), pp. 127–138.) via the Bessel function J _ν of the first kind, the Gauss hypergeometric function ₂ F ₁, the generalized hypergeometric function _p F _q and the Fox–Wright generalization _p Ψ _q of the hypergeometric function _p F _q , a number of integral representations of the Laplace, Fourier, and Mellin types are derived here for certain general families of Mathieu type series. Some interesting corollaries and consequences of these integral representations are also considered.

Keywords:

• integral representations
• Mathieu series
• alternating Mathieu series
• Laplace transforms
• Fourier sine transforms
• Fourier cosine transforms
• Mellin transforms
• Bessel function
• Gauss hypergeometric function
• generalized hypergeometric function
• Fox–Wright function
• Riemann Zeta function
• Dirichlet Eta function
• Sonine-Schafheitlin formula
• Weber-Sonine integral
• Legendre's duplication formula
• Gauss–Legendre multiplication formula

2000 Mathematics Subject Classifications :

• Primary: 33E20, 44A10
• Secondary: 33C10, 33C20, 44A20

Acknowledgements

The present investigation was supported, in part, by the Ministry of Education and Sciences of Macedonia and the Ministry of Education, Sciences and Sports of Croatia under Projects No. 05-437/1 and No. 13-272/1, and also, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.