On congruences related to central binomial coefficients, harmonic and Lucas numbers

Authors : Sibel KOPARAL, Neşe ÖMÜR

Pages : 973-985

View : 9 | Download : 7

Publication Date : 0000-00-00

Article Type : Research Paper

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}^{\leftinsert ignore into journalissuearticles values( p-1\right); /2}\leftinsert ignore into journalissuearticles values( -1\right); ^{k}\binom{2k}{k}% H_{k-1} &\equiv &\frac{2^{p}}{p}\leftinsert ignore into journalissuearticles values( 2F_{p+1}-5^{\leftinsert ignore into journalissuearticles values( p-1\right); /2}-1\right); insert ignore into journalissuearticles values({\rm mod\ }p);, \\ \sum\limits_{k=0}^{\leftinsert ignore into journalissuearticles values( p-1\right); /2}\frac{H_{k}C_{k}}{\leftinsert ignore into journalissuearticles values( -4\right); ^{k}} &\equiv &2\frac{Q_{p+1}}{p}-\frac{2^{p+1}}{p}\leftinsert ignore into journalissuearticles values( 1+2^{\leftinsert ignore into journalissuearticles values( p+1\right); /2}\right); insert ignore into journalissuearticles values({\rm mod\ }p);, \end{eqnarray*}% and for $\leftinsert ignore into journalissuearticles values( \frac{5}{p}\right); =1,$% \begin{equation*} \sum\limits_{k=1}^{\leftinsert ignore into journalissuearticles values( p-1\right); /2}\binom{2k}{k}\frac{H_{k-1}F_{k}}{% \leftinsert ignore into journalissuearticles values( -4\right); ^{k}}\equiv \frac{1}{p}\leftinsert ignore into journalissuearticles values( F_{2p+1}-F_{p+2}\right); -% \frac{2^{p}}{p}F_{p-1}insert ignore into journalissuearticles values({\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