Lucas-type congruences for cyclotomic $\ddot{I}ˆ$-coefficients

Abstract

Let p be any prime and a be a positive integer. For nonnegative integers l,n and an integer r, the normalized cyclotomic $\ddot{I}ˆ$-coefficient $${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-l{p}^a)/(p^{a-1}(p-1))]}\sum _{k=r(mod{p}^a)}(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k}\left)\right(\genfrac{}{}{0ex}{}{(k-r)/p^a}{l})$$ is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.