2026/07/08 by Zhi-Wei Sun
Mathematics · #math.NT #math.CO
For integers l>0 and m\geqslant0, we introduce the numbers Sl(m)(n) = ∑k1,…,kl∈\mathbb N\atop k1+⋯+kl = n \binom nk1,…,klm (n=0,1,2,…), and prove that for any prime p not dividing l+1 we have the congruence ∑n=1p-1\frac(-1)mnnm-1Sl(m)(n)≡0\pmod p. We also obtain a q-analogue of this result. For the Domb numbers given by D(n)=∑k=0n\binom nk2\binom2kk\binom2(n-k)n-k=S4(2)(n) (n=0,1,2,…), we confirm a previous conjecture which states that ∑n=1p-1\fracD(n)n≡(\frac p3)\frac 25pBp-2(\frac13)\pmodp2 for any prime p, where (\frac p3) is the Legendre symbol, and Bp-2(x) is the Bernoulli polynomial of degree p-2.