Combinatorial Sums $\sum_{k\equiv r(\mbox{mod } m)}{n\choose k}a^k$ and Lucas Quotients

TL;DR

This paper investigates specific combinatorial sums involving binomial coefficients and powers, deriving new congruences for Lucas quotients across two families of Lucas sequences, expanding known results beyond three sequences.

Contribution

It introduces novel congruences for Lucas quotients related to two infinite families of Lucas sequences, extending the scope of previous known results.

Findings

New congruences for Lucas quotients derived

Results apply to two infinite families of Lucas sequences

Extends known results beyond three sequences

Abstract

In this paper, we study the combinatorial sum $$\sum_{k\equiv r(\mbox{mod }m)}{n\choose k}a^k.$$ By studying this sum, we obtain new congruences for Lucas quotients of two infinite families of Lucas sequences. Only for three Lucas sequences, there are such known results.