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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 $p$.
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 fir
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjecture
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 i
In this paper, considering the Eichler-Shimura cohomology theory for Jacobi forms, we study connections between harmonic Maass-Jacobi forms and Jacobi integrals. As an application we study a pairing between two Jacobi integrals, which is defined by s