Raabe’s formula for p-adic gamma and zeta functions
Annales de l'Institut Fourier, Volume 58 (2008) no. 1, pp. 363-376.

The classical Raabe formula computes a definite integral of the logarithm of Euler’s Γ-function. We compute p-adic integrals of the p-adic logΓ-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.

La formule de Raabe classique donne la valeur de l’intégrale de la fonction log gamma d’Euler sur un intervalle de longueur 1. Nous calculons des intégrales p-adiques analogues pour les fonctions log gamma p-adiques de Diamond et de Morita, et nous montrons que chacune de ces fonctions est caractérisée de manière unique par son équation fonctionnelle et sa formule de Raabe p-adique. Nous démontrons aussi une formule de type Raabe pour les fonctions zêta de Hurwitz p-adiques.

DOI: 10.5802/aif.2353
Classification: 11S80, 11S40
Keywords: $p$-adic gamma function, $p$-adic zeta function, Raabe’s formula
Cohen, Henri 1; Friedman, Eduardo 2

1 Université Bordeaux I Institut de Mathématiques U.M.R. 5251 du C.N.R.S. 351 Cours de la Libération, 33405 Talence Cedex (France)
2 Universidad de Chile Facultad de Ciencias Departamento de Matemática Casilla 653 Santiago (Chile)
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Cohen, Henri; Friedman, Eduardo. Raabe’s formula for $p$-adic gamma and zeta functions. Annales de l'Institut Fourier, Volume 58 (2008) no. 1, pp. 363-376. doi : 10.5802/aif.2353. http://archive.numdam.org/articles/10.5802/aif.2353/

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