Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

Year:    2017

Analysis in Theory and Applications, Vol. 33 (2017), Iss. 4 : pp. 355–365

Abstract

In this paper some novel integrals associated with the product of classical Hermite’s polynomials $$\int^{+∞}_{−∞} (x^2)^m exp(−x^2)\{H_r(x)\}^2dx,\ \int^∞_0 exp(−x^2)H_{2k} (x)H_{2s+1}(x)dx,$$ $$\int^∞_0 exp(−x^2 )H_{2k}(x)H_{2s}(x)dx  \ \text{and}\ \int^∞_0 exp(−x^2)H_{2k+1}(x)H_{2s+1}(x)dx,$$ are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite’s polynomials by suitable simplifications of arbitrary parameters.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/ata.2017.v33.n4.5

Analysis in Theory and Applications, Vol. 33 (2017), Iss. 4 : pp. 355–365

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    11

Keywords:    Gauss’s summation theorem classical Hermite’s polynomials generalized hypergeometric function generalized Laguerre’s polynomials.