Notes on two kinds of special values for the Bell polynomials of the second kind

Abstract

In the paper, by methods and techniques in combinatorial analysis and the theory of special functions, the authors discuss two kinds of special values for the Bell polynomials of the second kind for two special sequences, find a relation between these two kinds of special values for the Bell polynomials of the second kind, and derive an identity involving the combinatorial numbers.

Special values of the Bell polynomials of the second kind for some sequences and functions

“…For details on discussion of these quantities, please refer to [70, Lemma 2.5], [75] and closely related references therein.…”

Three closed forms for convolved Fibonacci numbers

“…By the way, in the papers [8,20,34,39,43,48,54,50,55,56,59,64] and closely-related references therein, there are some new results and applications of special values of the Bell polynomials of the second kind B n,k . Remark 3.…”

Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions

“…In [14,Theorem 1] and [23, p. 80 for k ≥ 1 and n ≥ 0. The binomial inversion formula in [4, p. 192, (5.48) for k ≥ 1 and n ≥ 0, where The logarithmic derivative [ln (z)] = (z) (z) is denoted by ψ(z) and the derivatives ψ (k) (z) for k ≥ 0 are called polygamma functions.…”