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

Abstract

In this paper, with the help of the Faà di Bruno formula and an identity for the Bell polynomials of the second kind, we find several explicit formulas for the Peters polynomials and numbers. Also, we present determinantal representations for the Peters polynomials and numbers by virtue of a general derivative formula for the ratio of two differentiable functions. Moreover, we give several recursive relations for the Peters polynomials and numbers. As an application, we establish alternative recursive relations for them with the aid of a recursive relation for the Hessenberg determinants. These results cover counterparts for the Boole polynomials and numbers.

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.…”

“…Therefore, many researchers, including Qi et al have studied $$$ {B}_{n,k} $$$ to tackle some difficult issues and find effective results 20–26 . 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 27–31 …”

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

“…Proof. By using (26), we can easily furnish a proof of ( 27) and (28). Again, we write the generating function (24) in the following form…”

“…In order to prove our theorems, we give several lemmas below. For previous papers using this method, please see [25][26][27][28][29][30][31].…”