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

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.