Truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers (Q2414072)

\textit{L. Carlitz} introduced in [Duke Math. J. 1, 137--168 (1935; Zbl 0012.04904)] analogues of Bernoulli numbers for the global rational function field \({\mathbb F}_q(T)\), nowadays called Bernoulli-Carlitz numbers. The author and \textit{H. Kaneko} expressed explicitly in [J. Number Theory 163, 238--254 (2016; Zbl 1400.11065)] the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers by using the Sterling-Carlitz numbers of the second kind and of the first kind, respectively. This extends that the Bernoulli numbers and the Cauchy numbers are expressed explicitly by using the Stirling numbers of the second kind and of the first kind, respectively. In this paper, the author defines the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of the hypergeometric Bernoulli numbers and the hypergeometric Cauchy numbers, and as extensions of the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of the incomplete Stirling-Carlitz numbers.