A new recursive formula arising from a determinantal expression for weighted Delannoy numbers

On Degenerate Array Type Polynomials

“…Considering the generating function in (5) for x = 0, we proved the explicit formula (12). The proof of Theorem 3.1 is complete.…”

“…Remark 3.2. The explicit formula (12) in Theorem 3.1 and seven concrete values listed in Remark 3.1 reveal that degenerate λ-array type numbers S(n, m; 0; λ; γ ) are polynomials of λ and γ with degrees m and n − 1 ≥ 0, respectively. Theorem 3.2.…”

“…By virtue of the explicit formula (12), we obtain the first few values of degenerate λ-array type numbers S(n, m; 0; λ; γ ) for 1 ≤ n ≤ 6 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 generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials

“…[20][21][22][23][24][25][26] This approach has been also applied to the Bernoulli polynomials, the Euler polynomials, the generalized Eulerian polynomials, the Peters polynomials, and numbers that are involved in a significant position in mathematics to obtain meaningful relations and representations in previous papers. [27][28][29][30][31] In the present paper, by virtue of Faà di Bruno formula (2.1) in Lemma 2.1 and identities (2.2) and (2.3) in Lemmas 2.2 and 2.3 for the Bell polynomials of the second kind B n,k , we obtain an explicit formula for generalized Humbert-Hermite polynomials, in which falling and rising factorial are involved. We also derive a determinantal representation of generalized Humbert-Hermite polynomials H G (r) n+1,m (x, 𝑦, z) by using a general derivative formula (2.4) in Lemma 2 for the ratio of two differentiable functions.…”