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Frédéric Chapoton; Jiang Zeng
Nombres de $q$-Bernoulli–Carlitz et fractions continues
($q$-Bernoulli–Carlitz Numbers and continuous fractions)
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 347-368, doi: 10.5802/jtnb.984
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Class. Math.: 11B68, 30B70
Keywords: Nombre de Bernoulli, q-analogue, déterminant de Hankel, polynômes orthogonaux, fraction continue

Résumé - Abstract

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.

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