On Some Relations between the Hermite Polynomials and Some Well-Known Classical Polynomials and the Hypergeometric Function.

The connection between different classes of special functions is a very important aspect in establishing new properties of the related classical functions that is they can inherit the properties of each other. Here we show how the Hermite polynomials are related to some well-known classical polynomials such as the Legendre polynomials and the associated Laguerre polynomials. These relationships set up the connection between both kinds of classical orthogonal polynomials and grant us the ability to consider the theory of Hermite polynomials as a special case of the theory of Legendre and Laguerre polynomials. We show the confluent hypergeometric and the hypergeometric representations of Hermite polynomials. Thus the Hermite polynomials inherit the great advantage of carrying out their analytic continuation into any part of the complex z-plane. Furthermore, the hypergeometric representation enables us to develop the theory of the Hermite polynomials by implementing the general theory of the hypergeometric function. In this paper we have shown various types of formulae which link the Hermite polynomials to the Legendre polynomials. Some of these formulae are of integral form, operational form and an expansion form. Such diversity should grant us more flexibility to apply the Hermite polynomials in a variety of applications in mathematics, physics and engineering.