In this paper, we define the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers 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.
"Truncated Bernoulli-Carlitz and Truncated Cauchy-Carlitz Numbers." Tokyo J. Math. 41 (2) 541 - 556, December 2018. https://doi.org/10.3836/tjm/1502179245