The New Application of the Cauchy Operator

Abstract

In this paper, introduced new application of the Cauchy operator, where we define a polynomials ; , then we represent this polynomials by Cauchy operator, to derive the identities of ; in simple way, where we using this representation and the limit technique to introduce the basic identities: generating function, Mehler’s formula and Roger’s formula for ; polynomials. Also we introduce an extension of the generating function, extension of Mehler’s formula and extension of the Roger’s formula. All identities will be derived in this paper depending on the roles of the Cauchy operator given in [1,11].