Incomplete Hyperelliptic Integrals and Hypergeometric Series

Loiseau, J. F.; Codaccioni, J. P.; Caboz, R.

doi:10.2307/2008366

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\A < J2 4j L -1

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1996

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“…The function has also appeared in previous work, but its form has gone unnoticed. An example of this occurs in [4] where the calculation of roots of certain polynomial equations is considered. Here, the generalized binomial function appears in the form of a generalized hypergeometric function.…”

Section: \A < J2 4j L -mentioning

confidence: 99%

An Integral Representation for the Generalized Binomial Function

Heggie¹,

Nicklason²

1996

Can. math. bull.

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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.

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Section: \A < J2 4j L -mentioning

confidence: 99%