论文标题
一致性的二项式系数产物的一定数量
Congruences for certain lacunary sums of products of binomial coefficients
论文作者
论文摘要
结果表明，对于任何prime $ p $和任何自然数量$ \ ell，m，m，$和$ s $ m+s-1+\ ell（p-1）}＆\ equiv 0 \ bmod p \\ \ sum_ {i \ ge 0}（ - 1）^{m-i} {m \ select i} {\ ell+ip+ip+ip \ ell+ip \ eeld+ip \ select m+s-1}＆\ equiv 0 \ equiv 0 \ equiv 0 \ equ p^m \ \ sum_ \ ell}（-1）^{J-i} {m \ select J} {J \ select I} {J+S-1+I（P-1）\选择J+S-1+\ Ell（P-1）}＆Equiv 0 \ equiv 0 \ equiv 0 \ bmod P^{m- \ ell}相应的商涉及可以显式计算的Adelberg多项式，为这些总和提供封闭形式的表达式，即使$ p $不是Prime（当一致性不一定保留）时，即使是有效的。
It is shown that for any prime $p$ and any natural numbers $\ell, m,$ and $s$ such that $0<s<p$, the three following congruences \begin{align*}\sum_{i\ge \ell+1}(-1)^{m-i} {m \choose i}{m+s-1+i(p-1) \choose m+s-1+\ell(p-1)} &\equiv 0 \bmod p\\ \sum_{i\ge 0}(-1)^{m-i} {m \choose i}{\ell+ip \choose m+s-1}&\equiv 0 \bmod p^m\\ \sum_{j,i\ge \ell}(-1)^{j-i}{m \choose j} {j \choose i}{j+s-1+i(p-1) \choose j+s-1+\ell(p-1)}&\equiv 0 \bmod p^{m-\ell} \end{align*} hold true. The corresponding quotients involve Adelberg polynomials which can be computed explicitly, providing closed-form expressions for these sums, valid even if $p$ is not prime, when the congruences do not necessarily hold.
