A new kind of numbers and related congruences

Abstract

For integers $l>0$ and $m⩾0$, we introduce the numbers $$S_l^{(m)}(n)=\sum _{\genfrac{}{}{0ex}{}{k_1,\dots ,k_l\in \mathbb{N}}{k_1+\cdots +k_l=n}}{\left(\genfrac{}{}{0ex}{}{n}{k_1,\dots ,k_l}\right)}^m(n=0,1,2,\dots ),$$ and prove that for any prime $p$ not dividing $l+1$ we have the congruence $$\sum _{n=1}^{p-1}\frac{(-1{)}^{mn}}{n^{m-1}}{S}_l^{(m)}(n)\equiv 0(\mathrm{mod}p).$$ We also obtain a $q$-analogue of this result. For the Domb numbers given by $$D(n)=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\left(\genfrac{}{}{0ex}{}{2(n-k)}{n-k}\right)=S_4^{(2)}(n)(n=0,1,2,\dots ),$$ we confirm a previous conjecture which states that $$\sum _{n=1}^{p-1}\frac{D(n)}{n}\equiv \left(\frac{p}{3}\right)\frac{2}{5}p{B}_{p-2}\left(\frac{1}{3}\right)(\mathrm{mod}{p}^2)$$ for any prime $p$, where $(\frac{p}{3})$ is the Legendre symbol, and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

16 pages. Add a q-analogue of Theorem 1.1