MMN-1222
Some interesting congruences for ballot numbers
Nese Ömür; Sibel Koparal;Abstract
Let p be an odd prime. In this paper, we determine
∑_{k=0}^{p-1}(1/(m^{k}))((2k+d)/k)modp, ∑_{k=0}^{p-1}(k/(m^{k}))((2k+d)/k)modp
for d∈{0,1,...,p} and
∑_{k=0}^{p-1}((B(k,d))/(m^{k}))modp, ∑_{k=0}^{p-1}(k/(m^{k}))B(k,d)modp
for d∈{1,...,p}, where m∈ℤ with p∤mΔ. For example, for p≠5
∑_{k=0}^{p-1}(-1)^{k}((2k+d)/k)≡(-1)^{d+1}F_{d-2(p/5)}(modp),
and for p≠3
∑_{k=0}^{p-1}kB(k,d)≡(-1)^{d}d((⌊d/2⌋-p-(-1)^{d})/3)(modp),
where F_{n} is the nth Fibonacci number and (-) is the Jacobi symbol.
Vol. 19 (2018), No. 2, pp. 1079-1094