New congruences for sums involving Apery numbers or central Delannoy numbers

Details New congruences for sums involving Apery numbers or central Delannoy numbers New congruences for sums involving Apery numbers or central Delannoy numbers

2010-08-17

arxiv.org/abs/1008.2894v2

Mathematics

Number Theory

14 pages, submitted version, some conjectures are removed

Scientific paper

Let $r$ be a nonnegative integer and $n$ a positive integer. We obtain some new congruences involving the Ap\'ery numbers $A_n$ or central Delannoy numbers $D_n$ as follows: \sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &\equiv \sum_{k=0}^{n-1}\varepsilon^k (2k+1)^{2r+1}D_k \equiv 0\pmod n, where $\varepsilon=\pm 1$, $A_n=\sum_{k=0}^{n}{n+k\choose 2k}^2{2k\choose k}^2$, and $D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}$. These generalize several recent results of Z.-W. Sun. For $r=1$, we further prove that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where $p>3$ is a prime. Our proof depends heavily on the new congruence \sum_{k=0}^{n-1} {n+k\choose k}^2{n-1\choose k}^2 \equiv 0 \pmod{n}. We also derive some new congruences on sums of $q$-binomial coefficients.

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