Nombres de q-Bernoulli–Carlitz et fractions continues
[$q$-Bernoulli–Carlitz Numbers and continuous fractions]
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 347-368.

Carlitz introduced q-analogues of the Bernoulli numbers around 1950. We obtain a representation of these q-Bernoulli numbers (and some shifted version) as moments of some orthogonal polynomials. This also gives factorisations of Hankel determinants of q-Bernoulli numbers, and continued fractions for their generating series. Some of these results are q-analogues of known results for Bernoulli numbers, but some are specific to the q-Bernoulli setting.

Carlitz a introduit vers 1950 des q-analogues des nombres de Bernoulli. On obtient une représentation de ces q-analogues (ainsi que de variantes décalées) comme moments de certains polynômes orthogonaux. Ceci donne aussi des factorisations des déterminants de Hankel des nombres de q-Bernoulli, ainsi que des fractions continues pour leurs séries génératrices. Certains de ces résultats sont des q-analogues d’énoncés connus pour les nombres de Bernoulli, mais d’autres sont sans version classique.

Published online:
DOI: 10.5802/jtnb.984
Classification: 11B68,  30B70
Keywords: Nombre de Bernoulli, q-analogue, déterminant de Hankel, polynômes orthogonaux, fraction continue
     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},
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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.

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