A Study on Multi-Stirling Numbers of the First Kind

Abstract

In this paper, we define the multi-Stirling numbers of the first kind by means of the multiple logarithm and as a generalization of the Stirling numbers of the first kind. Then we introduce two additional special numbers by using the multiple logarithm, namely the modified multi-Bernoulli numbers as a generalization of the higher-order Bernoulli numbers and the multi-Lah numbers of type 2 as a generalization of the Lah numbers. For both of these special numbers, we derive explicit expressions for them which involve the multi-Stirling numbers of the first kind.

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

“…We recall that the falling factorial sequence is given by [1][2][3][4][5][7][8][9][10][11][12][13][14][15][16]18,19]). (1) It is well known that the Stirling numbers of the second kind are defined by…”