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Register a new userEvaluation of Some Sums Involving Powers of Harmonic 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 $tge 0$.
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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.
In recent years, the congruence $$ sum_{substack{i+j+k=p i,j,k>0}} frac1{ijk} equiv -2 B_{p-3} pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$.
We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.
Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.
In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.
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