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 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


INTRODUCTION
As is well known, the unsigned Stirling number n r counts the number of permutations of a set with n elements which are products of r disjoint cycles. We generalize these numbers to the multi-Stirling numbers of the first kind S (k 1 ,k 2 ,...,k r ) 1 (n, r) (see (10)) which reduce to the unsigned Stirling numbers of the first kind for (k 1 , k 2 , . . . , k r ) = (1, 1, . . . , 1). Indeed, S (1,1,...,1) 1 (n, r) = n r . It is also well known that the unsigned Lah numbers L(n, k) counts the number of ways of a set of n elements can be partitioned into k nonempty linearly ordered subsets. These numbers are generalized to the multi-Lah numbers L (k 1 ,k 2 ,...,k r ) (n, r) (see (18)) which reduce to the unsigned Lah numbers for (k 1 , k 2 , . . . , k r ) = (1, 1, . . . , 1). In fact, L (1,1,...,1) (n, r) = L(n, r).
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 the generalized Bernoulli numbers with 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) m m . 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 reduce to the poly-logarithm 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-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 Stirling numbers of the first kind are defined by , (see [5,8]), and the Stirling numbers of the second kind are defined by [2,3,8]).
The multi-Bernoulli numbers, which are called the generalized Bernoulli numbers with order r in [7], are defined by From (13), we note that where B , (see [1,7,8]).

MULTI-LAH NUMBERS AND MULTI-STIRLING NUMBERS OF THE FIRST KIND
Now, we define the multi-Stirling numbers of the first kind by Let us take k r = 1 in (11). Then we have d dt We claim that the following relations hold. For this, we only need to show the first equality which we prove by induction on r.
(13) Li 1, 1, . . . , 1 , as we wanted. Assume that r ≥ 2 and that it holds for r − 1. By (12) and induction hypothesis, we get Now, by (14) we obtain Thus our proof is completed. From (10), we note that where n r are the unsigned Stirling numbers of the first kind. We observe that (m, r) t n n! .