Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

On Degenerate Array Type Polynomials

“…Considering the generating function in (5), we proved the explicit formula (13). The proof of Theorem 3.2 is complete.…”

“…For new results and applications about the Bell polynomials' of the second kind B n,k , please refer to the papers [13,19,[21][22][23] and closely related references therein.…”

“…The proof of Theorem 3.2 is complete. By virtue of the explicit formula (13), we can calculate the first few values of degenerate λ-array type polynomials S(n, m; x; λ; γ ) for 1 ≤ n ≤ 3 as follows:…”

“…Let us notice that the Faá di Bruno formula, which can be viewed as an extension of chain rule to higher derivatives, has been applied to establish explicit and closed-form formulas of many important numbers and polynomials in analytic and combinatorial number theory. For more details, please refer to, for example, the papers [12][13][14][15][16][17][18] and closely related references therein.…”

Explicit, determinantal, recursive formulas and relations of the Peters polynomials and numbers

“…This method has been also applied to the Bernoulli and Euler polynomials and numbers, which are located in a very important position in mathematics so as to derive meaningful relations and representations in previous works. [25][26][27][28][29][30] In this paper, via the Faà di Bruno formula (2) in Lemma 2.1 and an identity (3) in Lemma 2.2 for the Bell polynomials of the second kind B n, k , we establish explicit formulas for the Peters polynomials and numbers, in which the falling and rising factorials ⟨x⟩ n and (x) n are involved. In addition, a determinantal representation of the Peters polynomials and numbers s n (x; 𝜆, 𝜇) and s n (𝜆, 𝜇) are deduced by making use of a general derivative formula (4) in Lemma 2.3 for the ratio of two differentiable functions.…”