A new generalization of the Stirling numbers of the first kind

In this paper, we study a generalization of the Stirling numbers of the first kind (classical and analogues versions) related to an extension of elementary symmetric function. These numbers appear as the coefficients of \(x^{k}\) $x^k$ in the expression \(\prod _{j=0}^{n-1}(x^{s}+jx^{s-1}+\cdots +j^{s-1}x+j^{s})\) ${\prod }_{j=0}^{n-1}(x^s+j{x}^{s-1}+\cdots +j^{s-1}x+j^s)$ . The array formed by these coefficients can be seen as a natural generalization of the first Stirling triangle. We also give a combinatorial interpretation of the classical numbers in terms of s-tuples of permutations of [n] with k cycles and using the inversion and co-inversion statistics on the cycles in the case of the analogues numbers, which also allows us to establish as a particular case the analogues of the Stirling numbers of the first kind. In addition, using the Legendre–Stirling numbers and the Stirling numbers of the first kind new formulas and useful properties are proposed.