The New Application of the Cauchy Operator

Mohammed A. Abdlhusein

College of Education for pure sciences, Thi-Qar University

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]. 

Key Words:  The Cauchy operator generating function , Mehler’s formula Rogers formula extended generating function extended Mehler’s formula 


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