Partial Lucas-type congruences

TL;DR

This paper generalizes Lucas-type congruences to sequences derived from constant terms of powers of multivariate Laurent polynomials, extending previous results related to binomial sums and Wolstenholme's theorem.

Contribution

It proves a broad theorem showing that such partial Lucas congruences hold for all sequences representable as constant terms of Laurent polynomial powers.

Findings

Sequences satisfy partial Lucas congruences under certain digit restrictions.

The result applies to a wide class of sequences including binomial sum related ones.

Extends previous Lucas-type congruence results to multivariate Laurent polynomial contexts.

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.