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 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 Families of Special Polynomials and Numbers Associated with Finite Operators

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 237
Author(s):  
Yilmaz Simsek
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Integral Representations ◽  
Euler Numbers ◽  
Derivative Operator ◽  
Riemann Integral ◽  
Special Polynomials ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Changhee Numbers

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

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 Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials

2020 ◽  
pp. 42-42
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Recurrence Formula ◽  
Binomial Coefficients ◽  
Combinatorial Identity ◽  
Lucas Numbers ◽  
Fundamental Properties ◽  
Special Polynomials ◽  
New Family ◽  
Special Numbers And Polynomials

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers 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 A Parametric Kind of the Degenerate Fubini Numbers and Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 405 ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Cheon Seoung Ryoo
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Trigonometric Functions ◽  
Summation Formulas ◽  
Degenerate Type

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling numbers.

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 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.

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