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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 2, June 2019  pp. 151–167.

On congruences involving special numbers

Authors:  Sibel Koparal and Nese Ömür
Author institution:Kocaeli University Mathematics Department 41380 Izmit Kocaeli Turkey

Summary:  In this paper, using some special numbers and combinatorial identities, we show some interesting congruences: for a prime p > 3,

∑ {k=0}{(p-1)/2} {C{k}}/{9{k}(k+1)} ≡ {2{p-1}}/{p}({L{2p}}/{3{p-2}}-9) -5 ({5}/{p}) +9 (mod p),

∑ {k=1}{(p-1)/2} ({2k}{k}) {H{k}{2}}/{3{k}} ≡ -{2}/{3} (-{1}/{3}){(p-1)/2} ({p}/{3}) B{p-2} ({1}/{3}) (mod p),

∑ {k=0}{(p-1)/2} ({2k}{k}) {F{k}}/{4{k}(2k+1)} ≡ {1}/{2p}(F{(1-3p)/2} - F{(1+3p)/2}) (mod p2),

where Bn(x) is the Bernoulli polynomial, Cn, Hn, Fn and Ln are the nth Catalan number, the nth harmonic number, the nth Fibonacci number and the nth Lucas number, respectively. (./p) denotes the Legendre symbol.

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