A Study on Some Properties of Legendre Polynomials and Integral Transforms

Abstract

The main aim of this paper is to use the integral representation method to derive the properties of Legendre polynomials. Their recurrence relations, differential equations, relationship with Hermite polynomials have been obtained.

Certain results for the Hermite and Chebyshev polynomials of 2-variables

“…The Hermite polynomials of the associated generating functions is reformulated within the framework of an operational formalism [5][6][7][8]. In the case of generalized special functions, the use of operational techniques, combined with the principle of monomiality [3,4,9] has provided new means of analysis for the derivation of the solution of large classes of partial differential equations often encountered in physical problems [10,14] offers a power tool to treat the relevant generating functions and the differential equations they satisfy. The results are interpreted in terms of single, several variables, single index, index two, three and in turn p-index in terms of Hermite polynomials defined by Srivastava [18,19].…”

On Horn Matrix Function <i>H<sub>2</sub>(A,A′,B,B′;C;z,w)</i> of Two Complex Variables under Differential Operator

“…In [6] [7] [8], the hypergeometric matrix function has been introduced as a matrix power series, an integral representation and the hypergeometric matrix differential equation. In [9]- [18], extension to the matrix function framework of the classical families of p-Kummers matrix function and Humbert matrix ⊂ Ω , then the properties of the matrix functional calculus [26], it follows that ( ) ( ) ( ) ( ). …”