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Register a new userFactors of certain sums involving central q-binomial coefficients
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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 conjectured by Guo. Our proof uses Ni and Pans technique and a simple $q$-congruence observed by Guo and Schlosser.
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We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate different families of power series which involve odd harmonic numbers and central binomial coefficients. Furthermore it is shown that based on these series a new type of nonlinear Euler sums that involve odd harmonic numbers can be calculated in terms of zeta functions.
We propose higher-order generalizations of Jacobsthals $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $binom{ip}{p}$ ($i=1,2,dots$) that are divisible by arbitrarily large powers of prime $p$.
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form [S_{alpha,beta}(n) := sum_{k,;ell}binom{2n}{n+k}binom{2n}{n+ell} |k^alpha-ell^alpha|^beta,] obtaining new results in the cases $alpha = 1, 2$. We show that there is a close connection between these double sums in the case $alpha=1$ and the single centered binomial sums considered by Tuenter.
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 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,winmathbb{N}={0,1,cdots})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3yin{4^k:kinmathbb{N}}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.
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