Incomplete Fibonacci and Lucas numbers (Q1916317)

It is well known that the Fibonacci numbers \(F_n\) and the Lucas numbers \(L_n\) can be written as \[ \begin{aligned} F_n &= \sum^k_{i=0} {{n-1-i} \choose i}, \qquad \lfloor (n- 1)/2 \rfloor\leq k\leq n-1, \tag{1}\\ L_n &= \sum^k_{i=0} {n\over {n-i}} {{n-i} \choose i}, \qquad \lfloor n/2 \rfloor \leq k\leq n-1. \tag{2} \end{aligned} \] The extended Fibonacci numbers \(G_n (k)\) and the extended Lucas numbers \(H_n (k)\) arise from (1) and (2) with \(k> n-1\), see the author, \textit{O. Brugia}, and \textit{A. F. Horadam} [Int. J. Math. Educ. Sci. Technol. 24, No. 1, 9-21 (1993; Zbl 0773.11014)] and the author, \textit{R. Meniocci}, and \textit{A. F. Horadam} [Fibonacci Q. 32, No. 5, 455-464 (1994; Zbl 0834.11008)]. In this paper the author studies the incomplete Fibonacci numbers \(F_n (k)\) arising from (1) with \(0\leq k\leq \lfloor (n- 1)/2 \rfloor\) and the incomplete Lucas numbers \(L_n (k)\) arising from (2) with \(0\leq k\leq \lfloor n/2 \rfloor\). The study of the congruence properties of \(L_n (k)\) leads to a new characterization of prime numbers.