The operator rϕs and the polynomials Kn

الملخص EN

Abstract Based on basic hypergeometric series, a new generalized -operator Φ has been constructed and obtained some operator identities.

Also, a new polynomial is introduced.

The generating function and its extension, Mehler’s formula and its extension and the Rogers formula for the polynomials have been achieved by using the operator Φ .

In fact, this work can be considered as a generalization of Liu work’s by imposing some special values of the parameters in our results.

Therefore the -Rogers-Szegö polynomials can be deduced directly.