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An Integral Representation for the Generalized Binomial Function

Published online by Cambridge University Press:  20 November 2018

M. Heggie  and
G. R. Nicklason
M. Heggie
Affiliation:
Sydney, Nova Scotia, B1P 6L2 e-mail:nicklason@caper2.uccb.ns.ca
G. R. Nicklason
Affiliation:
Sydney, Nova Scotia, B1P 6L2 e-mail:nicklason@caper2.uccb.ns.ca
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Abstract

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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.

Keywords

30E2033C20contour integral representationanalytic continuationgeneralized hypergeometric functionMellin-Barnes integral
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 59 - 67
Copyright
Copyright © Canadian Mathematical Society 1995

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