On Generating Functions for Boole Type Polynomials and Numbers of Higher Order and Their Applications

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.

“…Observe that

B_{n} (x) = \lim_{ω \to 1} B_{n} (x; ω) .

Here, B n ( x ) denotes the Bernoulli polynomials (cf. previous studies 1‐5,7‐33,34‐65,66,67 ; see also the references cited therein).…”

Section: Introductionmentioning

confidence: 60%

Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

Kucukoglu

2022

“…By applying the generating function methods, Simsek 9 established several novel identities, inequalities, and relations for the Peters type polynomials and for some combinatorial numbers and polynomials. For further studies, the reader can consult previous works [7][8][9][10][11][12][13][14][15][16][17][18] and plenty of references cited therein.…”

Section: Introductionmentioning

confidence: 99%

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

Dağli

2021

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.

“…8,12 , p. 128, 11 ) See, for detail, also (cf. [13][14][15][16][17][18][19][20][21][22][23] ). Setting x = 0 in (5), we have the Peters numbers, s n ( , ) = s n (0; , ).…”

Section: Introductionmentioning

confidence: 99%

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

Şimşek

2019

Self Cite

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p-adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well-known special numbers and polynomials are presented. By using p-adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol-type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well-known formulas. Finally, two open problems for interpolation functions for Apostol-type Peters numbers and polynomials are revealed.