Multi-Lah numbers and multi-Stirling numbers of the first kind

TL;DR

This paper introduces multi-Lah, multi-Stirling, and multi-Bernoulli numbers, explores their interrelations via generating functions involving multiple logarithms, and derives a recurrence relation for multi-Lah numbers.

Contribution

It presents new definitions of multi-Lah and multi-Stirling numbers of the first kind and establishes their relations with multi-Bernoulli numbers using generating functions.

Findings

Representation of multi-Bernoulli numbers via multi-Stirling numbers

Expression of multi-Lah numbers in terms of multi-Stirling numbers

A new recurrence relation for multi-Lah numbers

Abstract

In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to study several relations among those three numbers. In more detail, we represent the multi-Bernoulli numbers in terms of the multi-Stirling numbers of the first kind and vice versa, and the multi-Lah numbers in terms of multi-Stirling numbers. In addition, we deduce a recurrence relation for multi-Lah numbers