论文标题
与双重序列和加泰罗尼亚数字有关的一致性
Congruences Related to Dual Sequences and Catalan Numbers
论文作者
论文摘要
在对双序列的研究中，Sun引入了多项式\ [d_n（x，y）= \ sum_ {k = 0}^{n} {n} {n \ select k} {x \ select k} y^k} y^k \ text {and} s_n（x，y）= \ sum_ {k = 0}^{n} \ binom {n} {k} {k} \ binom {x} {k} {k} \ binom {-1-x} {k} {k} y^k。 \]已经建立了许多相关的一致性。在这里，我们通过确定\ [\ sum_ {k = 0}^{p-1} d_k（x_1，y_1）d_k（x_2，y_2，y_2）\ pmod p \ pmod p \ text {and} \ sum_ {k = 0} = 0}^{p-1} s_k（y_1，y_1，y_1，y_1，p p pmod pp p \ pm p p \ pm p p \ pm p p \ pm pm_1，y_1，y_1，对于任何奇数prime $ p $和$ p $ -adic Integers $ x_i，\ y_i $，in \ {1,2 \} $。考虑二项式系数和加泰罗尼亚数字之间的直接连接，我们还表征\ [\ sum_ {n = 0}^{p-1} \ left（\ sum_ {k = 0}^n {n {n \ select k} \ k} \ frac {c_k} {c_k} $ k $ th catalan编号，$ a \ in \ mathbb {z} \ setMinus \ {0 \} $，with $ \ gcd（a，p）= 1 $。这些确认并概括了太阳的一些猜想。
During the study of dual sequences, Sun introduced the polynomials \[ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. \] Many related congruences have been established and conjectured by Sun. Here we generalize some of them by determining \[ \sum_{k=0}^{p-1}D_k(x_1,y_1)D_k(x_2,y_2)\pmod p \text{ and } \sum_{k=0}^{p-1}S_k(x_1,y_1)S_k(x_2,y_2)\pmod p \] for any odd prime $p$ and $p$-adic integers $x_i,\ y_i$ with $i\in\{1,2\}$. Considering the immediate connection between binomial coefficients and Catalan numbers, we also characterize \[ \sum_{n=0}^{p-1}\left(\sum_{k=0}^n {n \choose k} \frac{C_k}{a^k}\right)^2 \pmod {p}, \] where $C_k$ denotes the $k$th Catalan number, $a\in\mathbb{Z}\setminus \{0\}$ with $\gcd(a,p)=1$. These confirm and generalise some of Sun's conjectures.
