Bernoulli-Carlitz and Cauchy-Carlitz numbers with Stirling-Carlitz numbers

2017/04/24 by Kaneko, Hajime, Komatsu, Takao
#05A15 #05A19 #11B68 #11B73 #11B75 #11R58 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1704.07135

Abstract

Recently, the Cauchy-Carlitz number was defined as the counterpart of the Bernoulli-Carlitz number. Both numbers can be expressed explicitly in terms of so-called Stirling-Carlitz numbers. In this paper, we study the second analogue of Stirling-Carlitz numbers and give some general formulae, including Bernoulli and Cauchy numbers in formal power series with complex coefficients, and Bernoulli-Carlitz and Cauchy-Carlitz numbers in function fields. We also give some applications of Hasse-Teichmüller derivative to hypergeometric Bernoulli and Cauchy numbers in terms of associated Stirling numbers.

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