8,453 research outputs found
On two conjectural supercongruences of Apagodu and Zeilberger
Let the numbers and denote \begin{align*}
\alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose
k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose
k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime
and positive integer \begin{align*} \alpha_{np}&\equiv
\left(\frac{p}{3}\right) \alpha_n \pmod{p^2},\\ \beta_{np}&\equiv \begin{cases}
\displaystyle \beta_n \pmod{p^2},\quad &\text{if },\\
-\gamma_n \pmod{p^2},\quad &\text{if }, \end{cases}
\end{align*} where denotes the Legendre symbol.
These two supercongruences were recently conjectured by Apagodu and Zeilberger.Comment: to appear in J. Difference Equ. Appl. This version is a bit different
from the final version for publicatio
Proof of some divisibility results on sums involving binomial coefficients
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four
supercongruences on sums involving binomial coefficients, which were originally
conjectured by Sun. We also confirm a related conjecture of Guo on
integer-valued polynomials.Comment: 6 page
On van Hamme's (A.2) and (H.2) supercongruences
In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled
(A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we
extend (A.2) and (H.2) to supercongruences modulo for primes , which appear to be new.Comment: 9 page
Congruences on sums of super Catalan numbers
In this paper, we prove two congruences on the double sums of the super
Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.Comment: 8 page
Supercongruences for the th Ap\'ery number
In this paper, we prove two conjectural supercongruences on the th
Ap\'ery number, which were recently proposed by Z.-H. Sun.Comment: 9 page
Proof of a congruence on sums of powers of -binomial coefficients
We prove that, if and are nonnegative
integers, then \begin{align*}
\frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack
a_i} \equiv 0\pmod{[n]}, \end{align*} where ,
, and . The case confirms
a recent conjecture of Z.-W. Sun. We also show that, if is a
prime, then \begin{align*} \frac{[a+b+1]!}{[a]![b]!}\sum_{h=0}^{p-1}q^h{h\brack
a}{h\brack b} \equiv (-1)^{a-b} q^{ab-{a\choose 2}-{b\choose
2}}[p]\pmod{[p]^2}. \end{align*}Comment: 5 page
Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum
The numbers and are defined as \begin{align*}
R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\
W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We
prove that, for any positive integer and odd prime , there hold
\begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\
\sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\
9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2
&\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if .}
\end{align*} The first two congruences were originally conjectured by Z.-W.
Sun. Our proof is based on the multi-variable Zeilberger algorithm and the
following observation: where .Comment: 18 page
-Analogues of two Ramanujan-type formulas for
We give -analogues of the following two Ramanujan-type formulas for
: \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3
4^k} =\frac{4}{\pi} \quad\text{and}\quad \sum_{k=0}^\infty
(-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 8^k } =\frac{2\sqrt{2}}{\pi}.
\end{align*} Our proof is based on two -WZ pairs found by the first author
in his earlier work.Comment: typos corrected, 5 page
Some supercongruences arising from symbolic summation
Based on some combinatorial identities arising from symbolic summation, we
extend two supercongruences on partial sums of hypergeometric series, which
were originally conjectured by Guo and Schlosser and recently confirmed by Jana
and Kalita.Comment: 11 page
On the divisibility of -trinomial coefficients
We establish a congruence on sums of central -binomial coefficients. From
this -congruence, we derive the divisibility of the -trinomial
coefficients introduced by Andrews and Baxter.Comment: 8 page
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