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

Jiangshuai Yang; Yingpu Deng

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arXiv · 1609.05409

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

Jiangshuai Yang·Yingpu Deng

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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.

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Jiangshuai Yang, Yingpu Deng. 2016-09-18. Combinatorial Sums $\sum_{k\equiv r(\mbox{mod} m)}{n\choose k}a^k$ and Lucas Quotients (II). https://arxiv.org/abs/1609.05409

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