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Henri Cohen; Eduardo Friedman
Raabe’s formula for $p$-adic gamma and zeta functions
(Formules de Raabe pour les fonctions gamma et zêta $p$-adiques)
Annales de l'institut Fourier, 58 no. 1 (2008), p. 363-376, doi: 10.5802/aif.2353
Article PDF | Reviews MR 2401225 | Zbl pre05267528
Class. Math.: 11S80, 11S40
Keywords: $p$-adic gamma function, $p$-adic zeta function, Raabe’s formula

Résumé - Abstract

The classical Raabe formula computes a definite integral of the logarithm of Euler’s $\Gamma $-function. We compute $p$-adic integrals of the $p$-adic $\log \Gamma $-functions, both Diamond’s and Morita’s, and show that each of these functions is uniquely characterized by its difference equation and $p$-adic Raabe formula. We also prove a Raabe-type formula for $p$-adic Hurwitz zeta functions.

Bibliography

[1] G. Andrews, R. Askey & R. Roy, Special Functions, Cambridge University Press, Cambridge, 2000  MR 1688958 |  Zbl 1075.33500
[2] J. Diamond, “The $p$-adic log gamma function and $p$-adic Euler constants”, Trans. Amer. Math. Soc. 233 (1977), p. 321-337  MR 498503 |  Zbl 0382.12008
[3] E. Friedman & S. N. M. Ruijsenaars, “Shintani-Barnes zeta and gamma functions”, Adv. in Math. 187 (2004), p. 362-395 Article |  MR 2078341 |  Zbl 02105047
[4] Y. Morita, “A $p$-adic analogue of the ${\Gamma } $-function”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), p. 255-266  MR 424762 |  Zbl 0308.12003
[5] Y. Morita, “On the Hurwitz-Lerch ${L}$-functions”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), p. 29-43  MR 441924 |  Zbl 0356.12019
[6] N. Nielsen, Handbuch der Theorie der Gammafunktion, Chelsea, New York, 1965, reprint of 1906 edition  JFM 37.0450.01
[7] A. Robert, A Course in $p$-adic Analysis, Springer-Verlag, Berlin, 2000  MR 1760253 |  Zbl 0947.11035
[8] W. H. Schikhof, An Introduction to Ultrametric Calculus, Cambridge, Cambridge University Press, 1984  MR 791759 |  Zbl 0553.26006
[9] L. Washington, “A note on $p$-adic ${L}$-functions”, J. Number Theory 8 (1976), p. 245-250 Article |  MR 406982 |  Zbl 0329.12017
[10] L. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, Berlin, 1982  MR 718674 |  Zbl 0484.12001
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