86.32 Sums of Powers of the Terms in Any Finite Arithmetic Progression

2002 ◽  
Vol 86 (506) ◽  
pp. 269 ◽  
Author(s):  
Martin Griffiths
2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si

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. 


2021 ◽  
Vol 27 (2) ◽  
pp. 101-110
Author(s):  
José Luis Cereceda

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.


2018 ◽  
Vol 10 (2) ◽  
pp. 5
Author(s):  
Do Tan Si

We prove that all the Faulhaber coefficients of a sum of odd power of elements of an arithmetic progression may simply be calculated from only one of them which is easily calculable from two Bernoulli polynomials as so as from power sums of integers. This gives two simple formulae for calculating them. As for sums related to even powers, they may be calculated simply from those related to the nearest odd one’s.


2014 ◽  
Vol 57 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Daniel M. Kane ◽  
Scott Duke Kominers

AbstractFor relatively prime positive integers u0 and r, we consider the least common multiple Ln := lcm(u0, u1..., un) of the finite arithmetic progression . We derive new lower bounds on Ln that improve upon those obtained previously when either u0 or n is large. When r is prime, our best bound is sharp up to a factor of n + 1 for u0 properly chosen, and is also nearly sharp as n → ∞.


Author(s):  
Sid Ali Bousla

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], we have [Formula: see text] where [Formula: see text]. If in addition [Formula: see text] is a prime number and [Formula: see text], then we prove that for any [Formula: see text], we have [Formula: see text], where [Formula: see text]. Finally, we apply those inequalities to estimate the arithmetic function [Formula: see text] defined by [Formula: see text] ([Formula: see text]), as well as some values of the generalized Chebyshev function [Formula: see text].


2017 ◽  
Vol 177 ◽  
pp. 37-42 ◽  
Author(s):  
Victor J.W. Guo ◽  
Qiang-Qiang Jiang

2013 ◽  
Vol 29 (4) ◽  
pp. 1211-1238 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Xavier Xarles

Sign in / Sign up

Export Citation Format

Share Document