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

Sibel KOPARAL; Neşe ÖMÜR

Makale detayları

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

Yazarlar:

Sibel KOPARAL | Neşe ÖMÜR

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2016 Vol.40 Issue 5, pp.973-985

Belge Tipi:

Makale

Yayın Tarihi:

9/2016

Belge Dili:

Türkçe

Document ID (Org No.):

2000746123

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Özet:

 

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 ppp, 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*}\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*}\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 (5p)=1,(5p)=1,\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*}\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*}\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 {Fn}{Fn}\left\{ F_{n}\right\}  is the Fibonacci sequence and {Qn}{Qn}\left\{  Q_{n}\right\}  is the Pell-Lucas sequence.

Makale detayları

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

Yazarlar: Sibel KOPARAL | Neşe ÖMÜR

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2016 Vol.40 Issue 5, pp.973-985

Belge Tipi: Makale

Yayın Tarihi: 9/2016

Belge Dili: Türkçe

Document ID (Org No.): 2000746123

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Sayı

Yazarlar:

Sibel KOPARAL | Neşe ÖMÜR

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2016 Vol.40 Issue 5, pp.973-985

Belge Tipi:

Makale

Yayın Tarihi:

9/2016

Belge Dili:

Türkçe

Document ID (Org No.):

2000746123

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098