Integral Representations of Generalized Mathieu Series Via Mittag-Leffler Type Functions

Mathematics Subject Classification: 33C05, 33C10, 33C20, 33C60, 33E12, 33E20, 40A30The main purpose of this paper is to present a number of potentially useful integral representations for the generalized Mathieu series as well as for its alternating versions via Mittag-Leffler type functions

On Hankel Transform of Generalized Mathieu Series

Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20By using integral representations for several Mathieu type series, a number of integral transforms of Hankel type are derived here for general families of Mathieu type series. These results generalize the corresponding ones on the Fourier transforms of Mathieu type series, obtained recently by Elezovic et al. [4], Tomovski [19] and Tomovski and Vu Kim Tuan [20]

Effects of a fractional friction with power-law memory kernel on string vibrations

AbstractIn this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power-law memory kernel. The exact solution is obtained in terms of the Mittag-Leffler type functions and a generalized integral operator containing a four parameter Mittag-Leffler function in the kernel. The method of separation of the variables, Laplace transform method and Sturm–Liouville problem are used to solve the equation analytically. The asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the Tauberian theorems. The proposed model may be used for describing processes where the memory effects of complex media could not be neglected

Weighted Hardy-type inequalities involving fractional calculus operators

The aim of this paper is to give a new class of general weighted Hardy-type inequalities involving an arbitrary convex function with some applications of generalized fractional calculus convolutive operators which contain Gauss-hypergeometric function, generalized Mittag-Leffler function and Hilfer fractional derivative operator, in the kernel

Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations

Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-P\"oschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.Comment: LaTeX2e with amsmath, amssymb, amscd 26 pages, no figure

New Upper Bounds for Mathieu-Type Series

The Mathieu’s series S(r) was considered firstly by É.L. Mathieu in 1890; its alternating variant Š(r) has been recently introduced by Pogány et al. [Some families of Mathieu a-series and alternating Mathieu a-series, 2006] where various bounds have been established for S, Š. In this note we obtain new upper bounds over S(r), Š(r) with the help of Hardy–Hilbert double integral inequality