scholarly journals Identities and relations for special numbers and polynomials: An approach to trigonometric functions

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 535-542
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Euler Numbers ◽  
Derived Relations ◽  
Special Numbers And Polynomials

In this paper, by using trigonometric functions and generating functions, identities and relations associated with special numbers and polynomials are derived. Relations among the combinatorial numbers, the Bernoulli polynomials, the Euler numbers, the Stirling numbers and others special numbers and polynomials are given.

scholarly journals Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

2020 ◽  
Vol 108 (122) ◽  
pp. 103-120
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Sums Of Powers ◽  
Alternating Sums ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Historical Remarks ◽  
Positive Integers

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

scholarly journals Degenerate Bell polynomials associated with umbral calculus

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han-Young Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernoulli Polynomials ◽  
Umbral Calculus ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Type Formula ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials

Abstract Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

scholarly journals Combinatorial identities and sums for special numbers and polynomials

Filomat ◽  
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.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
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.

scholarly journals A note on characteristic function for Bernstein polynomials involving special numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 543-549
Author(s):  
Buket Simsek
Keyword(s):  
Characteristic Function ◽  
Binomial Distribution ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Discrete Random Variable ◽  
Euler Numbers ◽  
Negative Order ◽  
Special Numbers And Polynomials

The aim of this present paper is to establish and study generating function associated with a characteristic function for the Bernstein polynomials. By this function, we derive many identities, relations and formulas relevant to moments of discrete random variable for the Bernstein polynomials (binomial distribution), Bernoulli numbers of negative order, Euler numbers of negative order and the Stirling numbers.

scholarly journals Generalized Tepper’s Identity and Its Application

Mathematics ◽  
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.

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

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 ◽  
Bernoulli Numbers And Polynomials ◽  
Array Polynomials ◽  
Euler Numbers And 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.

scholarly journals On the generalized q-poly-Euler polynomials of the second kind

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 475-482
Author(s):  
Veli Kurt
Keyword(s):  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Basic Properties

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

2021 ◽  
pp. 15-15
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Discrete Mathematics ◽  
New Class ◽  
Integral Formulas ◽  
Euler Numbers And Polynomials ◽  
Derivative Formulas ◽  
Special Numbers And Polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

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