Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Generalized Derivative ◽

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

On generalized q-poly-Bernoulli numbers and polynomials

Filomat ◽

10.2298/fil2002515b ◽

2020 ◽

Vol 34 (2) ◽

pp. 515-520

Author(s):

Secil Bilgic ◽

Veli Kurt

Keyword(s):

Bernoulli Polynomials ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Many mathematicians in ([1],[2],[5],[14],[20]) introduced and investigated the generalized q-Bernoulli numbers and polynomials and the generalized q-Euler numbers and polynomials and the generalized q-Gennochi numbers and polynomials. Mahmudov ([15],[16]) considered and investigated the q-Bernoulli polynomials B(?)n,q(x,y) in x,y of order ? and the q-Euler polynomials E(?) n,q (x,y)in x,y of order ?. In this work, we define generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ?. We give new relations between the generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ? and the generalized q-poly-Euler polynomials ?[k,?] n,q (x,y) in x,y of order ? and the q-Stirling numbers of the second kind S2,q(n,k).

Generalized Tepper’s Identity and Its Application

Mathematics ◽

10.3390/math8020243 ◽

2020 ◽

Vol 8 (2) ◽

pp. 243

Author(s):

Dmitry Kruchinin ◽

Vladimir Kruchinin ◽

Yilmaz Simsek

Keyword(s):

Number Theory ◽

Generating Functions ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Combinatorial Analysis ◽

Euler Numbers ◽

The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. Moreover, we give applications related to the Tepper identity and these numbers and polynomials.

Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach

Publications de l Institut Mathematique ◽

10.2298/pim1920113k ◽

2019 ◽

Vol 106 (120) ◽

pp. 113-123

Author(s):

Neslihan Kilar ◽

Yilmaz Simsek

Keyword(s):

Functional Equation ◽

Generating Functions ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Combinatorial Analysis ◽

Integral Approach ◽

Euler Numbers ◽

Array Polynomials ◽

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah numbers.

On (ρ,q)-Euler numbers and polynomials associated with (ρ,q)-Volkenborn integrals

International Journal of Number Theory ◽

10.1142/s179304211850015x ◽

2017 ◽

Vol 14 (01) ◽

pp. 241-253 ◽

Cited By ~ 1

Author(s):

Ugur Duran ◽

Mehmet Acikgoz

Keyword(s):

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Type Formula ◽

Euler Numbers And Polynomials ◽

Euler Polynomials And Numbers ◽

Haar Distribution

Recently, Araci et al. introduced [Formula: see text]-analogue of the Haar distribution and by means of the distribution, they constructed [Formula: see text]-Volkenborn integral yielding to Carlitz-type [Formula: see text]-Bernoulli numbers and polynomials. The aim of the present paper is to introduce a generalization of the fermionic [Formula: see text]-adic measure based on [Formula: see text]-integers and set the corresponding integral to this measure. Consequently, Carlitz-type [Formula: see text]-Euler polynomials and numbers are defined in terms of the above mentioned integral. Moreover, some of their identities and properties are established.

Simplifying Coefficients in Differential Equations Associated with Higher Order Bernoulli Numbers of the Second Kind

10.20944/preprints201708.0026.v1 ◽

2017 ◽

Cited By ~ 2

Author(s):

Feng Qi ◽

Da-Wei Niu ◽

Bai-Ni Guo

Keyword(s):

Differential Equations ◽

Inversion Formula ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Higher Order ◽

Bell Polynomials

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius–Euler Numbers

10.20944/preprints201708.0017.v1 ◽

2017 ◽

Author(s):

Feng Qi ◽

Da-Wei Niu ◽

Bai-Ni Guo

Keyword(s):

Differential Equations ◽

Stirling Numbers ◽

Higher Order ◽

Bell Polynomials ◽

Euler Numbers ◽

Inversion Formulas

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius-Euler Numbers

Tatra Mountains Mathematical Publications ◽

10.2478/tmmp-2018-0022 ◽

2018 ◽

Vol 72 (1) ◽

pp. 67-76 ◽

Cited By ~ 2

Author(s):

Feng Qi ◽

Da-Wei Niu ◽

Bai-Ni Guo

Keyword(s):

Differential Equations ◽

Stirling Numbers ◽

Higher Order ◽

Bell Polynomials ◽

Euler Numbers ◽

Inversion Formulas ◽

Faà Di Bruno Formula

Abstract In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

10.20944/preprints202007.0202.v1 ◽

2020 ◽

Author(s):

Waseem Khan

Keyword(s):

Arithmetic Function ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Polylogarithm Function ◽

Euler Numbers And Polynomials ◽

Special Numbers And Polynomials ◽

Euler Polynomials And Numbers ◽

New Type ◽

Logarithm Function

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

Symmetry ◽

10.3390/sym12010009 ◽

2019 ◽

Vol 12 (1) ◽

pp. 9 ◽

Cited By ~ 1

Author(s):

Daeyeoul Kim ◽

Yilmaz Simsek ◽

Ji Suk So

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Numbers ◽

Integral Formulas ◽

Euler Numbers And Polynomials ◽

Negative Order ◽

Changhee Polynomials ◽

Changhee Numbers

The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.

Ordinary and degenerate Euler numbers and polynomials

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2221-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Han Young Kim ◽

Jongkyum Kwon

Keyword(s):

Stirling Numbers ◽

Recursive Formula ◽

Euler Polynomials ◽

Euler Numbers ◽

Integer Power ◽

Power Sums ◽

Power Sum ◽

Abstract In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on $\mathbb{Z}_{p}$ Z p . Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.