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.

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.

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 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 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 New families of special numbers for computing negative order Euler numbers and related numbers and polynomials

2018 ◽  
Vol 12 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Euler Numbers ◽  
Open Questions ◽  
Negative Order ◽  
Central Factorial Numbers ◽  
Probability And Statistics

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

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

Symmetry ◽  
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.

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.

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