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

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.

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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 ◽

Bernoulli Numbers And Polynomials ◽

Euler Numbers And Polynomials

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.

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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.

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Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials

10.20944/preprints201610.0043.v1 ◽

2016 ◽

Cited By ~ 7

Author(s):

Feng Qi ◽

Bai-Ni Guo

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Polynomials ◽

Euler Numbers ◽

Generalized Derivative ◽

Bernoulli Numbers And Polynomials ◽

Euler Numbers And Polynomials

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.

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New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

Symmetry ◽

10.3390/sym13081484 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1484

Author(s):

Yilmaz Simsek

Keyword(s):

Functional Equation ◽

Generating Functions ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Polynomial Identities ◽

Euler Numbers ◽

Special Polynomial ◽

Combinatorial Sums ◽

Changhee Numbers ◽

Euler’S Identity

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

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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 ◽

Bernoulli Numbers And Polynomials ◽

Euler Numbers And Polynomials

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).

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Identities related to special polynomials and combinatorial numbers

Filomat ◽

10.2298/fil1715833y ◽

2017 ◽

Vol 31 (15) ◽

pp. 4833-4844 ◽

Cited By ~ 2

Author(s):

Eda Yuluklu ◽

Yilmaz Simsek ◽

Takao Komatsu

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Euler Numbers ◽

Special Polynomials ◽

Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

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A New Class of Degenerate Hermite Poly-Genocchi Polynomials

10.20944/preprints201807.0361.v1 ◽

2018 ◽

Author(s):

Waseem A. Khan ◽

K.S. Nisar

Keyword(s):

Generating Functions ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

General Symmetry ◽

Bernoulli Numbers And Polynomials ◽

New Class ◽

Genocchi Polynomials ◽

Summation Formulae

In this article, authors introduce a new class of degenerate Hermite poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials studied by Khan [8].

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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 ◽

Bernoulli Numbers And Polynomials ◽

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.

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Combinatorial identities and sums for special numbers and polynomials

Filomat ◽

10.2298/fil1820869s ◽

2018 ◽

Vol 32 (20) ◽

pp. 6869-6877

Author(s):

Yilmaz Simsek

Keyword(s):

Partial Differential Equations ◽

Generating Functions ◽

Functional Equations ◽

Stirling Numbers ◽

Euler Numbers ◽

Array Polynomials ◽

Central Factorial Numbers ◽

Special Numbers And Polynomials ◽

Combinatorial Sums ◽

Computation Algorithms

In this paper, by using some families of special numbers and polynomials with their generating functions and functional equations, we derive many new identities and relations related to these numbers and polynomials. These results are associated with well-known numbers and polynomials such as Euler numbers, Stirling numbers of the second kind, central factorial numbers and array polynomials. Furthermore, by using higher-order partial differential equations, we derive some combinatorial sums and identities. Finally, we give two computation algorithms for Euler numbers and central factorial numbers.

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New type of degenerate Daehee polynomials of the second kind

Advances in Difference Equations ◽

10.1186/s13662-020-02891-8 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Sunil Kumar Sharma ◽

Waseem A. Khan ◽

Serkan Araci ◽

Sameh S. Ahmed

Keyword(s):

Generating Function ◽

Special Class ◽

Stirling Numbers ◽

Bernoulli Numbers ◽

Higher Order ◽

Bernoulli Numbers And Polynomials ◽

New Type ◽

Order Degenerate ◽

Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

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