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 E(b theta). A new polynomials hn(a, b, x|q(-1)) are defined as an extension of the q(-1)-Rogers-Szegopolynomial h(n)(a, b|q(-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(n)(a, b|q(-1)). The generating function and its extension, Rogers formula and the invers linearization formula, and Mehler's formula for the polynomials h(n)(a, b|q(-1)) are deduced by giving special values to parameters of a new polynomial h(n)(a, b, x|q(-1)).