In this paper, we introduce the deformed homogeneous polynomials R₍ (x, y;u|q). These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials h₍ (x|q), the generalized Rogers-Szegö polynomials r₍ (x, y), the Stieltjes-Wigert polynomials S₍ (x;q), among others. Basic properties of the polynomial R₍ are given, along with recurrence relations, its q-difference equation, and representations. Generating functions for the polynomials R₍ (x, y;u|q) are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the q-binomial formula and the Heine transformation formula are obtained. These results are obtained via the u-deformed q-exponential operator E (yDₐ|u), defined here. From this operator, we obtain for free the operators T (yDₐ) the Chen, R (yDₐ) of Saad, E (yDₐ) of Exton, and R (yDₐ) of Rogers-Ramanujan when u=1, q, q, q², respectively. We introduce the deformed basic hypergeometric series ₑΦₒ, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.
Ronald Orozco López (Tue,) studied this question.