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

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 kinds of 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.

In addition, we need to recall the multi-Bernoulli numbers B (k 1 ,k 2 ,...,k r ) n (see (9)) which were introduced earlier under the different name of generalized Bernoulli numbers of order r in [7]. These numbers reduce to the Bernoulli numbers of order r up to some constants. Indeed, we see that B (1,1,...,1) The common feature of those three kinds of numbers is that they are all defined with the help of the multiple logarithm Li k 1 ,k 2 ,...,k r (z) (see (8)), which reduces to the polylogarithm Li k 1 (z), for r = 1.
The aim of this paper is to study several relations among those three kinds of 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-© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Stirling numbers. Moreover, we deduce a recurrence relation for multi-Lah numbers. For the rest of this section, we recall the necessary facts that will be needed throughout this paper.
The unsigned Stirling numbers L(n, k) are defined by The inverse formula of (1) is given by From (1), we can derive the generating function of unsigned Lah numbers given by Thus, we note that The Stirling numbers of the first kind are defined by and the Stirling numbers of the second kind are defined by From (4) and (5), we note that and For any k i ≥ 1 (1 ≤ i ≤ r), and |z| < 1, the multiple logarithm is defined by Li k 1 ,k 2 ,...,k r (z) = 0<m 1 <m 2 <···<m r z m r m k 1 1 m k 2 2 · · · m k r r see [1,7] .
The multi-Bernoulli numbers, which are called the generalized Bernoulli numbers of order r in [7], are defined by From (13), we note that where B (r) n are the Bernoulli numbers of order r given by

Multi-Lah numbers and multi-Stirling numbers of the first kind
Now, we define the multi-Stirling numbers of the first kind by where k i ≥ 1 (1 ≤ i ≤ r -1), k r ≥ 2, and |t| < 1.
We claim that the following relations hold. For this, we only need to show the first equality which we prove by induction on r: If r = 1, then Li 1 (t) = ∞ m=1 t m m = -log(1t), as we wanted. Assume that r ≥ 2 and that the relationship holds for r -1. By (12) and induction hypothesis, we get Now, by (14) Thus our proof is completed. From (10), we note that Therefore, by (13) and (16), we obtain the following lemma. Therefore, by (9) and (17), we obtain the following theorem. For any integer k i (i = 1, 2, . . . , r), in the view of (9), we define L (k 1 ,k 2 ,...,k r ) (n, r) for n, r ≥ 0, which are called multi-Lah numbers, as From (13), we note that  (1e -t ) m r +m r-1 (m r + m r-1 ) -k r From (20), we note that (-1) m r +l m r !m k r -j r S 2 (l, m r ) Therefore, by comparing the coefficients on both sides of (21), we obtain the following theorem.

Conclusion
There are various ways of studying special polynomials and numbers which include generating functions, combinatorial methods, p-adic analysis, umbral calculus, special functions, differential equations, and probability theory. In this paper, using the generating function method and by making use of the multiple logarithm, we studied three kinds of numbers, namely the multi-Stirling numbers of the first kind, the multi-Lah numbers, and the multi-Bernoulli numbers, which reduce respectively to the unsigned Stirling numbers of the first kind, the Lah numbers, and the higher-order Bernoulli numbers up to constants when the index is specialized to (k 1 , k 2 , . . . , k r ) = (1, 1, . . . , 1). We deduced several relations among those numbers. In more detail, we expressed 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. Further, we derived a recurrence relation for multi-Lah numbers. It is our continuous interest to explore some special numbers and polynomials by using different tools like those mentioned above.