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Special values of the hypergeometric series

Published online by Cambridge University Press:  24 October 2008

G. S. Joyce  and
I. J. Zucker
G. S. Joyce
Affiliation:
Wheatstone Physics Laboratory, King's College, Strand, London WC2R 2LS
I. J. Zucker
Affiliation:
Wheatstone Physics Laboratory, King's College, Strand, London WC2R 2LS
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Extract

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric series

for rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such as

and

where Γ(x) denotes the gamma function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 257 - 261
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Beukers, F. and Wolfart, J.. Algebraic values of hypergeometric functions. In New Advances in Transcendence Theory (Cambridge University Press, 1988).Google Scholar
[2]Borwein, J. M. and Borwein, P. B.. Pi and the AGM (Wiley, 1987).Google Scholar
[3]Flach, M.. Periods and special values of the hypergeometric series. Math. Proc. Cambridge Philos. Soc. 106 (1989), 389–401.CrossRefGoogle Scholar
[4]Glasser, M. L. and Wood, V. E.. A closed-form evaluation of the elliptic integral. Math. Comp. 25 (1971), 535–536.CrossRefGoogle Scholar
[5]Goursat, E.. Sur l'équation différentielle linéaire qui admet pour intégrale la serie hypergéométrique. Ann. Sci. École Norm. Sup. 10 (1881), S3–S142.CrossRefGoogle Scholar
[6]Kummer, E. E.. Über die hypergeometrische Reihe. J. Reine Angew. Math. 15 (1836), 39–83, 127–172.Google Scholar
[7]Selberg, A. and Chowla, S.. On Epstein's zeta function. J. Reine Angew. Math. 227 (1967), 86–110.Google Scholar
[8]Whittaker, E. T. and Watson, G. N.. A Course of Modern Analysis (Cambridge University Press, 1927).Google Scholar
[9]Wolfart, J.. Werte hypergeometrischer Funktionen. Invent. Math. 92 (1988), 187–216.Google Scholar
[10]Zucker, I. J.. The evaluation in terms of Γ-functions of the periods of elliptic curves admitting complex multiplication. Math. Proc. Cambridge Philos. Soc. 82 (1977), 111–118.CrossRefGoogle Scholar
[11]Zucker, I. J. and Robertson, M. M.. Further aspects of the evaluation of . Math. Proc. Cambridge Philos. Soc. 95 (1984), 5–13.CrossRefGoogle Scholar