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Turkish Journal of Mathematics

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.

DOI

10.3906/mat-1504-16

Keywords

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

First Page

973

Last Page

985

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