Partial Lucas-type congruences
Armin Straub
TL;DR
The paper addresses Lucas-type congruences for sequences by recasting them as constant terms $A(n)=\operatorname{ct}[P(\boldsymbol{x})^n]$ of multivariate Laurent polynomials. It establishes a general theorem: if $\frac{1}{M}\operatorname{Newt}(P)$ contains no nonzero integral interior points, then for every prime $p$, $A(p\,n+k)\equiv A(n)A(k)\pmod{p}$ whenever $k<\frac{p}{M}$, and it demonstrates this with concrete examples such as $u(n)=\operatorname{ct}[P(x)^n]$ for $P(x)=(1+x)^2(1-1/x)$ and its generalizations $u_{a,b}^{\varepsilon}(n)$. A refined congruence is derived for the upper digit range: $u(pn+k)\equiv w(n+1)u(k)\pmod{p}$ for $(p+1)/2\le k<p$, where $w(n)$ is a known binomial sum, with the Cartier operator playing a central role. The work connects these partial Lucas congruences to Apéry and Delannoy numbers, discusses potential converse statements about constant-term representations, and highlights open questions, including q-analogs and higher-power congruences.
Abstract
In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients with the notable twist that there is a restriction on the $p$-adic digits. We prove a general result that shows that similar partial Lucas congruences are satisfied by all sequences representable as the constant terms of the powers of a multivariate Laurent polynomial.