On introduit une déformation des séries de Dirichlet d’une variable complexe , sous la forme d’un opérateur pour chaque nombre complexe , agissant sur les séries formelles sans terme constant en une variable . On montre que les fractions de Bernoulli-Carlitz sont les images de certains polynômes en par les opérateurs associés à la fonction de Riemann aux entiers négatifs.
Bernoulli-Carlitz fractions and -Zeta operators
We introduce a deformation of Dirichlet series of one complex variable , which is given for each complex number by an operator acting on formal power series without constant term in the variable . We prove that the Bernoulli-Carlitz fractions are obtained as the image of some simple polynomials in by the operators corresponding to the Riemann function at negative integers.
@article{JTNB_2010__22_3_575_0,
author = {Fr\'ed\'eric Chapoton},
title = {Fractions de {Bernoulli-Carlitz} et op\'erateurs $q${-Zeta}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {575--581},
publisher = {Universit\'e Bordeaux 1},
volume = {22},
number = {3},
year = {2010},
doi = {10.5802/jtnb.733},
mrnumber = {2769332},
zbl = {1267.11097},
language = {fr},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.733/}
}
TY - JOUR AU - Frédéric Chapoton TI - Fractions de Bernoulli-Carlitz et opérateurs $q$-Zeta JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 575 EP - 581 VL - 22 IS - 3 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.733/ DO - 10.5802/jtnb.733 LA - fr ID - JTNB_2010__22_3_575_0 ER -
%0 Journal Article %A Frédéric Chapoton %T Fractions de Bernoulli-Carlitz et opérateurs $q$-Zeta %J Journal de théorie des nombres de Bordeaux %D 2010 %P 575-581 %V 22 %N 3 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.733/ %R 10.5802/jtnb.733 %G fr %F JTNB_2010__22_3_575_0
Frédéric Chapoton. Fractions de Bernoulli-Carlitz et opérateurs $q$-Zeta. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 575-581. doi : 10.5802/jtnb.733. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.733/
[1] -Bernoulli numbers and polynomials, Duke Math. J., Volume 15 (1948), pp. 987-1000 | MR | Zbl
[2] -Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., Volume 76 (1954), pp. 332-350 | MR | Zbl
[3] A rooted-trees -series lifting a one-parameter family of Lie idempotents, Algebra Number Theory, Volume 3 (2009) no. 6, pp. 611-636 | DOI | MR | Zbl
[4] A variation of Euler’s approach to values of the Riemann zeta function, Kyushu J. Math., Volume 57 (2003) no. 1, pp. 175-192 | MR | Zbl
[5] -Riemann zeta function, Int. J. Math. Math. Sci. (2004) no. 9-12, pp. 599-605 | MR | Zbl
[6] On Carlitz’s -Bernoulli numbers, J. Number Theory, Volume 14 (1982) no. 3, pp. 332-339 | MR | Zbl
[7] A spectral interpretation for the zeros of the Riemann zeta function, Mathematisches Institut, Georg-August-Universität Göttingen : Seminars Winter Term 2004/2005, Universitätsdrucke Göttingen, Göttingen, 2005, pp. 117-137 | MR | Zbl
[8] -analogue of Riemann’s -function and -Euler numbers, J. Number Theory, Volume 31 (1989) no. 3, pp. 346-362 | MR | Zbl
[9] A note on -analogues of the Dirichlet series and -Bernoulli numbers, J. Number Theory, Volume 39 (1991) no. 3, pp. 251-256 | MR | Zbl
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