On Mathieu-Type Series with (p,ν)-Extended Hypergeometric Terms: Integral Representations and Upper Bounds

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 S̃(r) has been recently introduced by Pogány et al. [12] where various bounds have been established for S, S̃. In this note we obtain new upper bounds over S(r), S̃(r) with the help of Hardy–Hilbert double integral inequality. 2000 Mathematics Subject Classification. Primary: 26D15, 33E20.

Integral expressions for Mathieu-type series whose terms contain Fox's H-function

Closed integral form expressions are derived for the Mathieu-type a-series and for the associated alternating series whose terms contain Fox’s H -function. In the final section, two similar results on Mathieu-type series with the Fox–Wright Psi-function pΨq and the generalized hypergeometric function pFq which can be expressed in terms of Fox’s H -functions are deduced respectively. c © 2006 El...

On Integral Forms of Generalised Mathieu Series

Integral representations for generalised Mathieu series are obtained which recapture the Mathieu series as a special case. Bounds are obtained through the use of the integral representations.

Multiple-integral Representations of Very-well-poised Hypergeometric Series

On hypergeometric series reductions from integral representations, the Kampé de Fériet function, and elsewhere

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrödinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to obtain analytic expressions, and we present the reduction of a number of cases of pFp and p+1Fp, mainly for p = 2 and p = 3. These and related series have addi...