Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers

Abstract

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.

Keywords

• bernoulli-carlitz numbers
• hypergeometric bernoulli numbers
• truncated bernoulli-carlitz numbers
• cauchy-carlitz numbers
• hypergeometric cauchy numbers
• truncated cauchy-carlitz numbers
• determinants
• recurrence relations
• function fields
• continued fractions