Reduction formulas for the summation of reciprocals in certain second-order recurring sequences (Q2785032)

\textit{Brother A. Brousseau} [Fibonacci Q. 7, 143-168 (1969; Zbl 0176.32203)] stated that the sums NEWLINE\[NEWLINE S(k_1, k_2,\ldots, k_m) =\sum_{n=1}^\infty {1\over F_nF_{n+k_1}F_{n+k_2}\cdots F_{n+k_m}} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE T(k_1, k_2,\ldots, k_m) =\sum_{n=1}^\infty {(-1)^{n-1}\over F_nF_{n+k_1}F_{n+k_2}\cdots F_{n+k_m}}, NEWLINE\]NEWLINE where \(\{F_n\}\) is the Fibonacci sequence, could be written as NEWLINE\[NEWLINE S(k_1, k_2,\ldots, k_m)=r_1+r_2 S(1, 2,\ldots, m) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE T(k_1, k_2,\ldots, k_m)=r_3+r_4 T(1, 2,\ldots, m), NEWLINE\]NEWLINE where \(r_1\), \(r_2\), \(r_3\), and \(r_4\) are rational numbers depending on \(k_1, k_2,\ldots, k_m\). \textit{R. André-Jeannin} [Fibonacci Q. 35, No. 1, 68-74 (1997; Zbl 0879.11006)] treated the case \(m=1\). The aim of this paper is to prove the claim for \(m\geq 2\). R. André-Jeannin and the present author, in fact, consider generalizations of the sums \(S(k_1, k_2,\ldots, k_m)\) and \(T(k_1, k_2,\ldots, k_m)\) replacing the Fibonacci sequence with Horadam's sequence [see \textit{A. F. Horadam}, Fibonacci Q. 3, 161-176 (1965; Zbl 0131.04103)].