The classical Raabe formula computes a definite integral of the logarithm of Euler’s -function. We compute -adic integrals of the -adic -functions, both Diamond’s and Morita’s, and show that each of these functions is uniquely characterized by its difference equation and -adic Raabe formula. We also prove a Raabe-type formula for -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 -adiques analogues pour les fonctions log gamma -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 -adique. Nous démontrons aussi une formule de type Raabe pour les fonctions zêta de Hurwitz -adiques.
Mots-clés : $p$-adic gamma function, $p$-adic zeta function, Raabe’s formula
@article{AIF_2008__58_1_363_0, author = {Cohen, Henri and Friedman, Eduardo}, title = {Raabe{\textquoteright}s formula for $p$-adic gamma and zeta functions}, journal = {Annales de l'Institut Fourier}, pages = {363--376}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {1}, year = {2008}, doi = {10.5802/aif.2353}, mrnumber = {2401225}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2353/} }
TY - JOUR AU - Cohen, Henri AU - Friedman, Eduardo TI - Raabe’s formula for $p$-adic gamma and zeta functions JO - Annales de l'Institut Fourier PY - 2008 SP - 363 EP - 376 VL - 58 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2353/ DO - 10.5802/aif.2353 LA - en ID - AIF_2008__58_1_363_0 ER -
%0 Journal Article %A Cohen, Henri %A Friedman, Eduardo %T Raabe’s formula for $p$-adic gamma and zeta functions %J Annales de l'Institut Fourier %D 2008 %P 363-376 %V 58 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2353/ %R 10.5802/aif.2353 %G en %F AIF_2008__58_1_363_0
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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