Deformed Homogeneous Polynomials and the Generalized q-Exponential Operator

In this paper, we introduce the deformed homogeneous polynomials Rₙ(x,y;u|q). These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials hₙ(x|q), the generalized Rogers-Szegö polynomials rₙ(x,y), the Stieltjes-Wigert polynomials Sₙ(x;q), among others. Basic properties of the polynomial Rₙ are given, along with recurrence relations, its q-difference equation, and representations. Generating functions for the polynomials Rₙ(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(yDq|u), defined here. From this operator, we obtain for free the operators T(yDq) the Chen, R(yDq) of Saad, E(yDq) of Exton, and R(yDq) of Rogers-Ramanujan when u=1,q,√q,q², respectively. We introduce the deformed basic hypergeometric series ᵣΦₛ, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.