THE HOMOGENEOUS q-DIFFERENCE OPERATOR AND THE RELATED POLYNOMIALS

We create the homogeneous q-difference operator [~.E](a, b; [theta]) as an extension of the exponential operator [epsilon](b[theta]). A new polynomials [h.sub.n](a, b, x|[q.sup.-1]) are defined as an extension of the [q.sup.-1]-Rogers-Szego polynomial [h.sub.n](a, b|[q.sup.-1]). We provide an operator proof of the generating function and its extension, Rogers formula and the invers linearization formula, and Mehler's formula for the polynomials [h.sub.n](a,b|[q.sup.-1]). The generating function and its extension, Rogers formula and the invers linearization formula, and Mehler's formula for the polynomials [h.sub.n](a, b|[q.sup.-1]) are deduced by giving special values to parameters of a new polynomial [h.sub.n](a, b, x|[q.sup.-1]). Keywords: the homogeneous q-difference operator, the [q.sup.-1]-Rogers-Szego polynomial, the generating function, the Rogers formula, the invers linearization formula, the Mehler's formula. AMS Subject Classification: 05A30, 33D45.