Generalized Ismail's argument and (f,g)-expansion formula

الأصل · EN

As further development of earlier works on the (f,g)-inversion, the present paper is devoted to the (f,g)-difference operator and the representation problem or an expansion formula of analytic functions. A recursive formula and the Leibniz formula for the (f,g)-difference operator of the product of two functions are established. The resulting expansion formula not only unifies the q-analogue of the Lagrange inversion formula of Gessel and Stanton (thus, a q-expansion formula of Liu) for q-series but also systematizes the "Ismail's argument". In the meantime, a rigorous analytic proof of the (1-xy,x-y)-expansion formula with respect to geometric series, along with a proof of the previously unknown fact that it is equivalent to a q-analogue of the Lagrange inversion formula due to Gessel and Stanton, is presented. As applications, new proofs of several well-known summation and transformation formulas are investigated.