The $(f,g)$-Difference Operator, $(f,g)$-Expansion Formula and Their Applications

X. Ma
Department of Mathematics, Suzhou University, SuZhou, P. R. China


Abstract     Full Text  PDF

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 function. A recursive formula and the Leibniz formula for the $(f,g)$-difference operator of the product of two functions are established. The corresponding 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 Gessel and Stanton's $q$-analogue is presented. As applications, new proofs of several well-known summation and transformation formulas are investigated.