Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers

2012 ◽  
Vol 285 (17-18) ◽  
pp. 2129-2156 ◽  
Author(s):  
Stamatis Koumandos ◽  
Henrik Laurberg Pedersen
Keyword(s):  
Generating Functions ◽  
Partial Sums ◽  
Euler Numbers ◽  
Turán Type Inequalities

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 Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

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 Multivariate Interpolation Functions of Higher-Orderq-Euler Numbers and Their Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Hacer Ozden ◽  
Ismail Naci Cangul ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Higher Order ◽  
Multivariate Interpolation ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Sums Of Products ◽  
Interpolation Functions

The aim of this paper, firstly, is to construct generating functions ofq-Euler numbers and polynomials of higher order by applying the fermionicp-adicq-Volkenborn integral, secondly, to define multivariateq-Euler zeta function (Barnes-type Hurwitzq-Euler zeta function) andl-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitzq-Euler zeta function and multivariateq-Eulerl-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.

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 Sums and Partial Sums of Horadam Sequences: The Sum Formulas of $\sum_{k=0}^{n}x^{k}W_{k}$ and $ \sum_{k=n}^{n+m}x^{k}W_{k}$ via Generating Functions

2021 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Partial Sums

In this paper, we present sums and partial sums of Horadam sequences via generating functions which extends a recent result of Prodinger.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

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