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

TL;DR

This paper explores congruences related to combinatorial sums involving binomial coefficients and Lucas quotients, expressing these sums through recurrent sequences and providing new congruence results.

Contribution

It introduces a novel method to express combinatorial sums in terms of recurrent sequences with bounded order, extending previous results on Lucas quotients.

Findings

Expressed combinatorial sums using recurrent sequences of order up to φ(m)

Derived new congruences for Lucas quotients

Extended previous results on Lucas sequences and sums

Abstract

In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed in terms of some recurrent sequences with orders not exceeding $\varphi{(m)}$ and give some new congruences.