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An Integral Representation for the Generalized Binomial Function
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- Journal:
- Canadian Mathematical Bulletin / Volume 39 / Issue 1 / 01 March 1996
- Published online by Cambridge University Press:
- 20 November 2018, pp. 59-67
- Print publication:
- 01 March 1996
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- Article
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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.
can be obtained as the solution of the equation 