Geodesic
An Integral Representation for the Generalized Binomial Function
Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 59-67

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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.
DOI : 10.4153/CMB-1996-008-4
Mots-clés : 30E20, 33C20, contour integral representation, analytic continuation, generalized hypergeometricfunction, Mellin-Barnes integral 
Heggie, M.; Nicklason, G. R. An Integral Representation for the Generalized Binomial Function. Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 59-67. doi: 10.4153/CMB-1996-008-4
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     journal = {Canadian mathematical bulletin},
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     year = {1996},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-008-4/}
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