The $(f,g)$-inversion formula and its applications: the $(f,g)$-summation formula

A complete characterization of two functions $f(x,y)$ and $g(x,y)$ in the $(f,g)$-inversion is presented. As an application to the theory of hypergeometric series, a general bibasic summation formula determined by $f(x,y)$ and $g(x,y)$ as well as four arbitrary sequences is obtained which unifies Gasper and Rahman’s, Chu’s and Macdonald’s bibasic summation formula. Furthermore, an alternative proof of the $(f,g)$-inversion derived from the $(f,g)$-summation formula is presented. A bilateral $(f,g)$-inversion containing Schlosser’s bilateral matrix inversion as a special case is also obtained.