Catalan-Larcombe-French数的超同余

Catalan-Larcombe-French数的同余

Congruences for Catalan-Larcombe-French numbers

令$\{P_n\}$是由$P_0=1,\ P_1=8$和$n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2}$ $(n\ge 2)$给出的Catalan-Larcombe-French数，并令$S_n=P_n/2^n$。本文给出了$S_{mp^r}\pmod{p^{r+2}}$，$S_{mp^r-1}\pmod{p^r}$和$S_{mp^r+1}\pmod{p^{2r}}$的同余式，其中$p$是奇素数，$m,r$是正整数.我们还证明了对任何素数$p\equiv 5,7\pmod 8$，都有$S_{(p^2-1)/2}\equiv 0\pmod {p^2}$，并证明了$\{S_m\}$是对数凸的。

涉及Franel数和Catalan-Larcombe-French数的同余

Congruences involving Franel and Catalan-Larcombe-French numbers

令$\{f_n\}$是由$f_n=\sum_{k=0}^n\binom nk^3$给出的弗兰纳数，令$p>5$是素数。本文主要是对$m=5,-16,16,32,-49,50,96$进行$\sum_{k=0}^{p-1} \binom{2k}k\frac{f_k}{m^k}\pmod p$的测定。令$S_n=\sum_{k=0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k}$。我们还确定了$m=7,16,25,32,64,160,800,1600, 156832$的$\sum_{k=0}^{p-1} \binom{2k}k\frac{S_k}{m^k}\pmod p$。

Z.-证明了两个超同余W.Sun涉及Catalan-Larcombe-French数

Proof of two supercongruences conjectured by Z.-W.Sun involving Catalan-Larcombe-French numbers

调和数$H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$在数学中起着重要的作用。借助于组合恒等式，我们建立了以下两个同余式：$$\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\ \mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\ \mbox{modulo}\ p$$对任意素数$p>3$，第二个同余式由Z. W. 2012年的太阳这两个同余对于证明孙志文的下列定理是非常重要的：对于任何旧素数$p$，我们有$$\sum_{k=0}^{p-1}\frac{P_k}{8^k}\equiv1+2\big(\frac{-1}p\big)p^2E_{p-3}\pmod{p^3}$$和$$\sum_{k=0}^{p-1}\frac{P_k}{{16}^k}\equiv\big(\frac{-1}p\big)-p^2E_{p-3}\pmod{p^3},$$，其中$P_n=\sum_{k=0}^n\frac{\binom{2k}k^2\binom{2(n-k)}{n-k}^2}{\binom nk}$是第n个Catalan-Larcombe-French数。

A supercongruence involving cubes of Catalan numbers

We mainly show a supercongruence for a truncated series with cubes of Catalan numbers which extends a result by Zhi-Wei Sun.

Proof of Sun's supercongruence involving Catalan numbers

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.