The Cauchy Operator and the Homogeneous Hahn Polynomials

The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ(α)n(x,y|q)).