scholarly journals Some Computational Formulas forD-Nörlund Numbers

2009 ◽  
Vol 2009 ◽  
pp. 1-7 ◽  
Author(s):  
Guodong Liu
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Central Factorial Numbers

The author establishes some identities involving theDnumbers, Bernoulli numbers, and central factorial numbers of the first kind. A generating function and several computational formulas forD-Nörlund numbers are also presented.

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 New type of degenerate Daehee polynomials of the second kind

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sunil Kumar Sharma ◽  
Waseem A. Khan ◽  
Serkan Araci ◽  
Sameh S. Ahmed
Keyword(s):  
Generating Function ◽  
Special Class ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials ◽  
New Type ◽  
Order Degenerate ◽  
Math Phys

Abstract Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

scholarly journals A note on p-adic Carlitz's q-Bernoulli numbers

2000 ◽  
Vol 62 (2) ◽  
pp. 227-234 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim
Keyword(s):  
Invariant Measure ◽  
Generating Function ◽  
Bernoulli Number ◽  
Bernoulli Numbers ◽  
Adic Case

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

scholarly journals Explicit Formulas Involving -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong Jin Seo
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Derivative Operator ◽  
Euler Numbers And Polynomials ◽  
Integer Ring

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

scholarly journals Some Properties of a Sequence Similar to Generalized Euler Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Haiqing Wang ◽  
Guodong Liu
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Central Factorial Numbers

We introduce the sequence {Un(x)} given by generating function (1/(et+e-t-1))x=∑n=0∞Un(x)(tn/n!)  (|t|<(1/3)π,1x:=1) and establish some explicit formulas for the sequence {Un(x)}. Several identities involving the sequence {Un(x)}, Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

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