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

Voir la notice de l'article provenant de la source Cambridge University Press

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
@article{10_4153_CMB_1996_008_4,
     author = {Heggie, M. and Nicklason, G. R.},
     title = {An {Integral} {Representation} for the {Generalized} {Binomial} {Function}},
     journal = {Canadian mathematical bulletin},
     pages = {59--67},
     year = {1996},
     volume = {39},
     number = {1},
     doi = {10.4153/CMB-1996-008-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-008-4/}
}
TY  - JOUR
AU  - Heggie, M.
AU  - Nicklason, G. R.
TI  - An Integral Representation for the Generalized Binomial Function
JO  - Canadian mathematical bulletin
PY  - 1996
SP  - 59
EP  - 67
VL  - 39
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-008-4/
DO  - 10.4153/CMB-1996-008-4
ID  - 10_4153_CMB_1996_008_4
ER  - 
%0 Journal Article
%A Heggie, M.
%A Nicklason, G. R.
%T An Integral Representation for the Generalized Binomial Function
%J Canadian mathematical bulletin
%D 1996
%P 59-67
%V 39
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-008-4/
%R 10.4153/CMB-1996-008-4
%F 10_4153_CMB_1996_008_4

[1] 1. Graham, R. L., Knuth, D. E. and Patashnik, O., Concrete mathematics, Addison-Wesley, New York (1989), 200. Google Scholar

[2] 2. Heggie, M. and Nicklason, G. R., Quadrature of an exact shock solution and the generalized binomial function; preprint. Google Scholar

[3] 3. Heggie, M. and Nicklason, G. R., The generalized binomial function and “singular hyperelliptic “ integrals; preprint. Google Scholar

[4] 4. Loiseau, J. F., Codaccioni, J. P. and R. Caboz, Incomplete hyperelliptic integrals and hypergeometric series, Math, of Comp. 53(1989), 335–342. Google Scholar

[5] 5. Nicklason, G. R. and Heggie, M., An exact shock solution, Can. Appl. Math. Quart. 1(1993), 221–254. Google Scholar

[6] 6. Nicklason, G. R. and Heggie, M., Application of the generalized binomial function to one-dimensional shock problems; preprint. Google Scholar

[7] 7. Rainville, E. D., Special functions, Chelsea, New York, 1971. Google Scholar

[8] 8. Whittaker, E. T. and Watson, G. N., A course in modern analysis, 4th éd., Cambridge Univ. Press, New York, 1927. Google Scholar

Cité par Sources :