APPLICATIONS OF THE GENERALIZED HOMOGENEOUS q-SHIFT OPERATOR IN q-POLYNOMIALS

APPLICATIONS OF THE GENERALIZED HOMOGENEOUS q-SHIFT OPERATOR IN q-POLYNOMIALS

S. A. Abdul-Ghani, H. L.

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Abstract

In this paper, we construct the generalized homogeneous q-shift operator r s   a1;       ; ar b1;       ; bs ; q; cDxy   . Then, we apply this operator to derive some q-identities such as: the generating function and its extension, Rogers formula and its extension, Mehler's formula and its extension, Srivastava-Agarwal type bilinear generating functions for the polynomials  (a;b) n (x; y; cjq). Also, we obtain a transformation formula involving generating functions for  (a;b) n (x; y; cjq). We provide some special values for  (a;b) n (x; y; cjq) in order to establish identities for the polynomials  (a) n (x) and  (a;b) n (x; yjq).

Keywords

the homogeneous q-di erence operator, the homogeneous q-shift operator, q- Hahn polynomials, the generalized Al-Salam-Carlitz q-polynomials, generating function, Rogers formula, Mehler's formula.