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Barnes' First Lemma and its Finite Analogue

Published online by Cambridge University Press:  20 November 2018

Anna Helversen-Pasotto  and
Patrick Solé
Anna Helversen-Pasotto
Affiliation:
U.N.S.A. Faculté des Sciences Département de Mathématiques CNRS-URA 168 Parc Valrose B.R 71 06108 NICE-CEDEX 2 France
Patrick Solé
Affiliation:
CNRS-URA 1376 Laboratoire 1.3.S, Bâtiment 4 250 rue Albert Einstein Sophia-Antipolis 06560 Valbonne France
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Abstract

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We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

Keywords

33A1511L0520C99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 273 - 282
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Bailey, W. N., Generalised Hypergeometric Series, Cambridge Math. Tract, 1935, second edition, 1964.Google Scholar
2. Cartan, H., Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, Hermann, 1961, sixth edition, 1978.Google Scholar
3. Erdelyi, A. et ai, Higher Transcendental Functions, II, McGraw-Hill, 1953, reprinted Krieger, R. E., 1985.Google Scholar
4. Evans, R., Identities for products of Gauss sums over finite fields, Ens. Math. 27(1981), 197–209.Google Scholar
5. Evgrafov, R. A., Analytic Functions, Dover, 1966.Google Scholar
6. Greene, J., Hyper geometric Functions over Finite Fields, Trans, of the AMS, (1) 301(1987), 77–101.Google Scholar
7. Helversen-Pasotto, A., Représentations de Gelfand-Graev et identités de Barnes: le cas de GL2 d'un corps fini, Ens. Math. 32(1986), 57–77. 8 , L'identité de Barnes pour les corps finis, C.R.A.S., Paris (A) 286(1978), 297–300.Google Scholar
9. Ireland, K. and Rosen, M., A Classical Introduction to Modem Number Theory, second edition, Springer GTM84 1990.Google Scholar
10. Jacobi, C., Uberdie Kreisteilung und ihre Anwendung aufdie Zahlentheorie, J. Reine Angew. Math. (1846), 166-182.Google Scholar
11. Katz, N. M., Gauss sums, Kloosterman sums and Monodromy Groups, Ann. of Math. Studies, 116, Princeton University Press, (1988).Google Scholar
12. Koblitz, N., The number of points of certain families of hypersurfaces over finite fields, Comp. Math. 48(1983), 3–23.Google Scholar
13. Li, W. W. and J. Soto-Andrade, Barnes ‘ identities and representations of GL(2). I. Finite Field Case, J. Reine und Angew. Math. 344(1983), 171–179.Google Scholar
14. Li, W. W., Barnes’ identities and representations of GL(2). II. Non Local Field Archimedian Case, J. Reine und Angew. Math. 345(1983), 69–92.Google Scholar
15. Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Ellis Horwood, 1983.Google Scholar
16. Slater, L. J., Generalized Hypergeometric Functions, Cambridge University Press, 1966.Google Scholar
17. Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis, Cambridge, 1958.Google Scholar