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# On congruences related to central binomial coefficients, harmonic and Lucasnumbers

## DOI

10.3906/mat-1504-16

## Abstract

In this paper, using some combinatorial identities, we present new congruences involving central binomial coefficients and harmonic, Catalan, and Fibonacci numbers. For example, for an odd prime $p$, we have \begin{eqnarray*} \sum\limits_{k=1}^{\left( p-1\right) /2}\left( -1\right) ^{k}\binom{2k}{k}% H_{k-1} &\equiv &\frac{2^{p}}{p}\left( 2F_{p+1}-5^{\left( p-1\right) /2}-1\right) ({\rm mod\ }p), \\ \sum\limits_{k=0}^{\left( p-1\right) /2}\frac{H_{k}C_{k}}{\left( -4\right) ^{k}} &\equiv &2\frac{Q_{p+1}}{p}-\frac{2^{p+1}}{p}\left( 1+2^{\left( p+1\right) /2}\right) ({\rm mod\ }p), \end{eqnarray*}% and for $\left( \frac{5}{p}\right) =1,$% \begin{equation*} \sum\limits_{k=1}^{\left( p-1\right) /2}\binom{2k}{k}\frac{H_{k-1}F_{k}}{% \left( -4\right) ^{k}}\equiv \frac{1}{p}\left( F_{2p+1}-F_{p+2}\right) -% \frac{2^{p}}{p}F_{p-1}({\rm mod\ }p), \end{equation*} where $\left\{ F_{n}\right\}$ is the Fibonacci sequence and $\left\{ Q_{n}\right\}$ is the Pell-Lucas sequence.

## Keywords

Central binomial coefficients, harmonic numbers, Catalan numbers, Fibonacci numbers, Pell numbers

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