Evaluación de algunas integrales que involucran los polinomios clásicos de Hermite y Legendre, usando el método de transformadas de Laplace y el enfoque hipergeométrico

• M. I. Qureshi Department of Applied Sciences and Humanities Faculty of Engineering and Technology Central University Jamia Millia Islamia, New Delhi-110025
• Saima Jabee
• M Shadab Department of Applied Sciences and Humanities Faculty of Engineering and Technology Central University Jamia Millia Islamia, New Delhi-110025.

Keywords: Gauss's summation theorem, classical Legendre's polynomials of first kind, classical Hermite's polynomials, generalized hypergeometric function, Laplace transformation

Abstract

In this paper we have described some novel integrals associated with different higher order polynomials such as classical Hermite's polynomials and classical Legendre's polynomials. The following integrals
\begin{equation*}
{\int_{-\infty}^{+\infty}{x^{n}}{\exp(-x^2)}{{H_{n-2k}(x)}}}dx~,
{\int_{-\infty}^{+\infty}{x^{k}}{\exp(-x^2)}{{H_{n}(x)}}}dx~,
\end{equation*}
\begin{equation*}
{\int_{0}^{\infty}{t^{n}}{\exp(-t^2)}{{H_{n}(xt)}}}dt ~~\text{ and }~
{\int_{x}^{\infty}{t^{n+1}}{\exp(-t^2)}{{P_{n}\left(\frac{x}{t}\right)}}}dt
\end{equation*}
are evaluated using hypergeometric approach and Laplace transform technique, which is a different approach from the approaches given by the other authors in the field of special functions.

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