An Integral Representation for the Generalized Binomial Function

Abstract

The generalized binomial function can be obtained as the solution of the equation y = 1 +zyα which satisfies y(0) = 1 where α ≠ 1 is assumed to be real and positive. The technique of Lagrange inversion can be used to express as a series which converges for |z| < α-α|a — l|α-1. We obtain a representation of the function as a contour integral and show that if α > 1 it is an analytic function in the complex z plane cut along the nonnegative real axis. For 0 < α < 1 the region of analyticity is the sector |arg(—z)| < απ. In either case defined by the series can be continued beyond the circle of convergenece of the series through a functional equation which can be derived from the integral representation.

Raney Distributions and Random Matrix Theory

“…x(w − 1) = w 1−g which is precisely (2.2) in the case r = 1, with z = x and p = 1 − g. Another point of interest is that the analytic function defined by the power series on the right hand side of (2.1) has been the subject of a number of earlier studies [37,38,6]. In particular, with B p (z) = (1/z)G p,1 (1/z), it is shown in [37] that…”