Abstract
In the paper, with the help of the Faá di Bruno formula and an identity of the Bell polynomials of the second kind, the authors define degenerate λ-array type polynomials, establish two explicit formulas, and present several recurrence relations of degenerate λ-array type polynomials and numbers.
Highlights
In this paper, we use the following notation:Z = {0, ±1, ±2, . . .}, N = {1, 2, . . .}, N0 = {0, 1, 2, . . .}, N− = {−1, −2, . . . }.The Stirling numbers of the second kind S(n, m) for n ≥ m ≥ 0 can be generated by et − 1 m = ∞ S (n, m) tn (1) m! n=m n!and can be computed asS(n, m) = 1 m (−1)i m (m − i)n. i i=0See [[1], p.206] and the paper [2]
In the paper [7], Bayad et al deduced interesting and meaningful identities associated with λ-array type polynomials, λ-Stirling numbers of the second kind, and the Apostol–Bernoulli numbers, while they dealt with λ-array polynomials by applying λ-delta operator
Considering the generating function in (5) for x = 0, we proved the explicit formula (12)
Summary
We use the following notation:. Z = {0, ±1, ±2, . . .}, N = {1, 2, . . .}, N0 = {0, 1, 2, . . .}, N− = {−1, −2, . . . }. The λ-array type polynomials S(n, m; x; λ) were defined in [3] by the generating function (λet − 1)m ext =. In the paper [7], Bayad et al deduced interesting and meaningful identities associated with λ-array type polynomials, λ-Stirling numbers of the second kind, and the Apostol–Bernoulli numbers, while they dealt with λ-array polynomials by applying λ-delta operator. Readers interested to the Apostol-Bernoulli numbers and polynomials may consult to the papers [8,9,10] and closely related references therein. (3) and (4) reduce to the generating functions for classical Bernoulli and Euler polynomials, respectively. In this paper, utilizing the Faá di Bruno formula and an identity of the Bell polynomials of the second kind, we establish several explicit formulas and recurrence relations of (degenerate) λ-array type numbers and polynomials. Please refer to, for example, the papers [12,13,14,15,16,17,18] and closely related references therein
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