Generalized Ismail's argument and $(f,g)$-expansion formula
Abstract
As further development of earlier works on the -inversion, the present paper is devoted to the -difference operator and the representation problem or an expansion formula of analytic functions. A recursive formula and the Leibniz formula for the -difference operator of the product of two functions are established. The resulting expansion formula not only unifies the -analogue of the Lagrange inversion formula of Gessel and Stanton (thus, a -expansion formula of Liu) for -series but also systematizes the "Ismail's argument". In the meantime, a rigorous analytic proof of the -expansion formula with respect to geometric series, along with a proof of the previously unknown fact that it is equivalent to a -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.
Cite
@article{arxiv.math/0608674,
title = {Generalized Ismail's argument and $(f,g)$-expansion formula},
author = {Xinrong Ma},
journal= {arXiv preprint arXiv:math/0608674},
year = {2007}
}
Comments
31 pages