[-Bernoulli–Carlitz Numbers and continuous fractions]
Carlitz introduced -analogues of the Bernoulli numbers around 1950. We obtain a representation of these -Bernoulli numbers (and some shifted version) as moments of some orthogonal polynomials. This also gives factorisations of Hankel determinants of -Bernoulli numbers, and continued fractions for their generating series. Some of these results are -analogues of known results for Bernoulli numbers, but some are specific to the -Bernoulli setting.
Carlitz a introduit vers 1950 des -analogues des nombres de Bernoulli. On obtient une représentation de ces -analogues (ainsi que de variantes décalées) comme moments de certains polynômes orthogonaux. Ceci donne aussi des factorisations des déterminants de Hankel des nombres de -Bernoulli, ainsi que des fractions continues pour leurs séries génératrices. Certains de ces résultats sont des -analogues d’énoncés connus pour les nombres de Bernoulli, mais d’autres sont sans version classique.
Revised:
Accepted:
Published online:
Keywords: Nombre de Bernoulli, q-analogue, déterminant de Hankel, polynômes orthogonaux, fraction continue
Frédéric Chapoton 1; Jiang Zeng 2
CC-BY-ND 4.0
@article{JTNB_2017__29_2_347_0,
author = {Fr\'ed\'eric Chapoton and Jiang Zeng},
title = {Nombres de $q${-Bernoulli{\textendash}Carlitz} et fractions continues},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {347--368},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {29},
number = {2},
year = {2017},
doi = {10.5802/jtnb.984},
language = {fr},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.984/}
}
TY - JOUR AU - Frédéric Chapoton AU - Jiang Zeng TI - Nombres de $q$-Bernoulli–Carlitz et fractions continues JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 347 EP - 368 VL - 29 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.984/ DO - 10.5802/jtnb.984 LA - fr ID - JTNB_2017__29_2_347_0 ER -
%0 Journal Article %A Frédéric Chapoton %A Jiang Zeng %T Nombres de $q$-Bernoulli–Carlitz et fractions continues %J Journal de théorie des nombres de Bordeaux %D 2017 %P 347-368 %V 29 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.984/ %R 10.5802/jtnb.984 %G fr %F JTNB_2017__29_2_347_0
Frédéric Chapoton; Jiang Zeng. Nombres de $q$-Bernoulli–Carlitz et fractions continues. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 347-368. doi : 10.5802/jtnb.984. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.984/
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