Raabe’s formula for p -adic gamma and zeta functions

Henri Cohen[1]; Eduardo Friedman[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)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 363-376
  • ISSN: 0373-0956

Abstract

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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 log Γ -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.

How to cite

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Cohen, Henri, and Friedman, Eduardo. "Raabe’s formula for $p$-adic gamma and zeta functions." Annales de l’institut Fourier 58.1 (2008): 363-376. <http://eudml.org/doc/10315>.

@article{Cohen2008,
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.},
affiliation = {Université Bordeaux I Institut de Mathématiques U.M.R. 5251 du C.N.R.S. 351 Cours de la Libération, 33405 Talence Cedex (France); Universidad de Chile Facultad de Ciencias Departamento de Matemática Casilla 653 Santiago (Chile)},
author = {Cohen, Henri, Friedman, Eduardo},
journal = {Annales de l’institut Fourier},
keywords = {$p$-adic gamma function; $p$-adic zeta function; Raabe’s formula; -adic gamma function; -adic zeta function; Raabe's formula},
language = {eng},
number = {1},
pages = {363-376},
publisher = {Association des Annales de l’institut Fourier},
title = {Raabe’s formula for $p$-adic gamma and zeta functions},
url = {http://eudml.org/doc/10315},
volume = {58},
year = {2008},
}

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
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 363
EP - 376
AB - 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.
LA - eng
KW - $p$-adic gamma function; $p$-adic zeta function; Raabe’s formula; -adic gamma function; -adic zeta function; Raabe's formula
UR - http://eudml.org/doc/10315
ER -

References

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  1. G. Andrews, R. Askey, R. Roy, Special Functions, (2000), Cambridge University Press, Cambridge Zbl1075.33500MR1688958
  2. J. Diamond, The p -adic log gamma function and p -adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321-337 Zbl0382.12008MR498503
  3. E. Friedman, S. N. M. Ruijsenaars, Shintani-Barnes zeta and gamma functions, Adv. in Math. 187 (2004), 362-395 Zbl1112.11042MR2078341
  4. Y. Morita, A p -adic analogue of the Γ -function, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 255-266 Zbl0308.12003MR424762
  5. Y. Morita, On the Hurwitz-Lerch L -functions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 29-43 Zbl0356.12019MR441924
  6. N. Nielsen, Handbuch der Theorie der Gammafunktion, (1965), Chelsea, New York 
  7. A. Robert, A Course in p -adic Analysis, (2000), Springer-Verlag, Berlin Zbl0947.11035MR1760253
  8. W. H. Schikhof, An Introduction to Ultrametric Calculus, (1984), Cambridge, Cambridge University Press Zbl0553.26006MR791759
  9. L. Washington, A note on p -adic L -functions, J. Number Theory 8 (1976), 245-250 Zbl0329.12017MR406982
  10. L. Washington, Introduction to Cyclotomic Fields, (1982), Springer-Verlag, Berlin Zbl0484.12001MR718674

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