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Register a new userBernoulli numbers and sums of powers of integers of higher order
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We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
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In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were first studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.
A set $mathcal{A}$ is said to be an additive $h$-basis if each element in ${0,1,ldots,hn}$ can be written as an $h$-sum of elements of $mathcal{A}$ in {it at least} one way. We seek multiple representations as $h$-sums, and, in this paper we make a start by restricting ourselves to $h=2$. We say that $mathcal{A}$ is said to be a truncated $(alpha,2,g)$ additive basis if each $jin[alpha n, (2-alpha)n]$ can be represented as a $2$-sum of elements of $mathcal{A}$ in at least $g$ ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated $(alpha,2,g)$ additive basis with high or low probability.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
In the past two decades, many researchers have studied {it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${mathbb Z}/m{mathbb Z}$ for the order $m$ of the multiplicative character involved. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields. On the other hand, Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004, 2012) studied {it pure} Gauss sums, some nonzero integral powers of which are in the field of rational numbers. In this paper, we study Gauss sums, some integral powers of which are in quadratic fields. This class of Gauss sums is a generalization of index $2$ Gauss sums and an extension of pure Gauss sums to quadratic fields.
The aim of this paper is to prove wordlessly the sum formula of $1^{k}+2^{k}+ldots +n^{k}$, $kin{1,2,3}$.
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