Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach (Analysis in Theory and Applications, 2017)
In this paper some novel integrals associated with the product of classicalHermite’s polynomials $$\int^{+∞}_{−∞} (x^2)^m exp(−x^2)\{H_r(x)\}^2dx,\ \int^∞_0exp(−x^2)H_{2k} (x)H_{2s+1}(x)dx,$$ $$\int^∞_0exp(−x^2)H_{2k}(x)H_{2s}(x)dx  \ \text{and}\ \int^∞_0exp(−x^2)H_{2k+1}(x)H_{2s+1}(x)dx,$$ are evaluated using hypergeometric approach and Laplace transform method, which isa 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 numerousother interesting integrals involving the product of classical Hermite’s polynomials bysuitable simplifications of arbitrary parameters.
Keywords: Gauss’s summation theorem; classical Hermite’s polynomials; generalized hypergeometric function; generalized Laguerre’s polynomials.