scholarly journals Sums of Products Involving Power Sums of φ(n) Integers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jitender Singh
Keyword(s):  
Closed Form ◽  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Rational Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Power Sums ◽  
Power Sum ◽  
Mobius Function ◽  
Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d,  n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

scholarly journals Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

scholarly journals On the Symmetries of theq-Bernoulli Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Reflection Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sum ◽  
Congruence Properties

Kupershmidt and Tuenter have introduced reflection symmetries for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.

DEFINING POWER SUMS OF n AND φ(n) INTEGERS

2009 ◽  
Vol 05 (01) ◽  
pp. 41-53 ◽  
Author(s):  
JITENDER SINGH
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Power Sums ◽  
Power Sum ◽  
Analytical Results ◽  
General Power ◽  
Integral Relations ◽  
Positive Integers

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

scholarly journals Arithmetical properties of double Möbius-Bernoulli numbers

2019 ◽  
Vol 17 (1) ◽  
pp. 32-42
Author(s):  
Abdelmejid Bayad ◽  
Daeyeoul Kim ◽  
Yan Li
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Critical Values ◽  
Möbius Function ◽  
Dedekind Sums ◽  
Mobius Function ◽  
Dirichlet Type ◽  
Positive Integers

Abstract Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers Mk(n), double Möbius-Bernoulli numbers Mk(n,n′), and Möbius-Bernoulli polynomials Mk(n)(x). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials Mk(n)(x), can be interpreted as critical values of the following Dirichlet type L-function $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.

scholarly journals On a multidimensional Volkenborn integral and higher order Bernoulli numbers

2002 ◽  
Vol 65 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Min-Soo Kim ◽  
Jin-Woo Son
Keyword(s):  
Bernoulli Numbers ◽  
Higher Order ◽  
Sums Of Products

In this paper, using a multidimensional Volkenborn integral, we give a p-adic expression of the higher order Bernoulli numbers. This shows immediately the relation to the sums of products of the ordinary Bernoulli numbers of Dilcher in 1996. We also consider the Mahler expansion of several p-adic variables function, and give some examples.

scholarly journals The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials

2020 ◽  
pp. 1-16
Author(s):  
Arnold Adelberg
Keyword(s):  
Previous Analysis ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bernoulli Numbers And Polynomials

Several new estimates for the [Formula: see text]-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of [Formula: see text] in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

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