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

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 352 ◽  
Author(s):  
Yilmaz Simsek ◽  
Ji So
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Functional Equations ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Euler Polynomials ◽  
Derivative Formulas ◽  
Combinatorial Sums

The purpose of this manuscript is to study and investigate generating functions for Boole type polynomials and numbers of higher order. With the help of these generating functions, many properties of Boole type polynomials and numbers are presented. By applications of partial derivative and functional equations for these functions, derivative formulas, recurrence relations and finite combinatorial sums involving the Apostol-Euler polynomials, the Stirling numbers and the Daehee numbers are given.

scholarly journals Some new identities and formulas for higher-order combinatorial-type numbers and polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 551-558
Author(s):  
Irem Kucukoglu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Combinatorial Type ◽  
Derivative Formulas ◽  
Cauchy Numbers ◽  
Special Numbers And Polynomials ◽  
Boole Polynomials ◽  
Changhee Numbers

The main purpose of this paper is to provide various identities and formulas for higherorder combinatorial-type numbers and polynomials with the help of generating functions and their both functional equations and derivative formulas. The results of this paper comprise some special numbers and polynomials such as the Stirling numbers of the first kind, the Cauchy numbers, the Changhee numbers, the Simsek numbers, the Peters poynomials, the Boole polynomials, the Simsek polynomials. Finally, remarks and observations on our results are given.

scholarly journals Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

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.

scholarly journals Formulas derived from moment generating functions and Bernstein polynomials

2019 ◽  
Vol 13 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Buket Simsek
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Functional Equations ◽  
Bernstein Polynomials ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Derivative Formula ◽  
Moment Generating Functions ◽  
Derivative Formulas

The purpose of this paper is to provide some identities derived by moment generating functions and characteristics functions. By using functional equations of the generating functions for the combinatorial numbers y1 (m,n,?), defined in [12, p. 8, Theorem 1], we obtain some new formulas for moments of discrete random variable that follows binomial (Newton) distribution with an application of the Bernstein polynomials. Finally, we present partial derivative formulas for moment generating functions which involve derivative formula of the Bernstein polynomials.

scholarly journals Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1097 ◽  
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Partial Derivative ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Derivative Operator ◽  
Fundamental Properties ◽  
Special Polynomials ◽  
Euler’S Formula ◽  
Combinatorial Sums

The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

scholarly journals Computation of k-ary Lyndon words using generating functions and their differential equations

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3455-3463
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Prime Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Combinatorial Sums ◽  
Higher Order Derivative ◽  
Lyndon Words ◽  
Higher Order Differential Equations

By using generating functions technique, we investigate some properties of the k-ary Lyndon words. We give an explicit formula for the generating functions including not only combinatorial sums, but also hypergeometric function. We also derive higher-order differential equations and some formulas related to the k-ary Lyndon words. By applying these equations and formulas, we also derive some novel identities including the Stirling numbers of the second kind, the Apostol-Bernoulli numbers and combinatorial sums. Moreover, in order to compute numerical values of the higher-order derivative for the generating functions enumerating k-ary Lyndon words with prime number length, we construct an efficient algorithm. By applying this algorithm, we give some numerical values for these derivative equations for selected different prime numbers.

scholarly journals On a family of special numbers and polynomials associated with Apostol-type numbers and polynomials and combinatorial numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 478-494 ◽  
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Special Numbers And Polynomials ◽  
Combinatorial Sums ◽  
Finite Sums ◽  
Computation Algorithms

In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including ?-Apostol-Daehee numbers, special numbers, and finite sums.

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.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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.

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