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arXiv · 2104.01746

Appell-Carlitz numbers

Su Hu·Min-Soo Kim
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Abstract

In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the definitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers.Their generating function is usually named Hurwitz series in the function field arithmetic. By using Hasse-Teichm\"uller derivatives, we also obtain several properties of the (higher order) Appell-Carlitz numbers, including a recurrence formula, two closed forms expressions, and a determinant expression. The recurrence formula implies Carlitz's recurrence formula for Bernoulli-Carlitz numbers. Two closed from expressions implies the corresponding results for Bernoulli-Carlitz and Cauchy-Carlitz numbers . The determinant expression implies the corresponding results for Bernoulli-Carlitz and Cauchy-Carlitz numbers, which are analogues of the classical determinant expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in 1875.

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Su Hu, Min-Soo Kim. 2021-04-05. Appell-Carlitz numbers. https://arxiv.org/abs/2104.01746

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