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

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.