On new identities for Bell's polynomials

Identities via Bell matrix and Fibonacci matrix

“…Particularly, we have B 0,0 = 1, B n,0 = 0, B n,1 = x n for n ≥ 1 and B n,k = 0 for n < k. In addition to these, the following lemma holds [7, p. 135]. The readers are referred to [1] and [4,Chapter 11] for some other sequences which can be obtained from the Bell polynomials. Now, define the n × n Bell matrix B n by (B n ) i, j = B i, j and denote (S n ) i, j = (F −1 n ) i, j = f i, j , where i, j = 1, 2, .…”

“…It is well known that many combinatorial sequences, for instance, the Stirling numbers and the Lah numbers, are special cases of the Bell polynomials (see [1], [4,Chapter 11] and [7,Chapter 3]). Therefore, by means of the study of the matrix related to the Bell polynomials, we will have a unified approach to various lower triangular matrices.…”

Cvijovic's addition formula for the Lah numbers

“…Therefore, by virtue of the relationship (7) between B n,k and L(n, k), and noting that B n,k = 0 for k > n, the identity (2) follows upon setting x i = i! in Cvijović's addition formula (1). Incidentally, it is worth noting that combining Equations (6) and (7) yields the following alternative definition of the Lah numbers L(n, k) =…”

On Matrix Identities and Determinant Identities for Composite Functions

“…Many sequences such as the Stirling numbers and the Lah numbers are special values of the Bell polynomials. Recently, Abbas and Bouroubi [1] proposed two new methods for determining new identities for the Bell polynomials, and Yang [20] generalized one of the methods. By studying the matrices related to the Bell polynomials, Wang and Wang [16] gave a unified approach to various lower triangular matrices such as the Stirling matrices of the first kind and of the second kind [5,6], the Lah matrix [14] and so on.…”