Reciprocal sums of second-order recurrent sequences (Q2735217)

There exist a lot of results for the calculation of reciprocal sums of a second-order recurrent sequence. For the Fibonacci sequence, I. J. Good and V. E. Hoggatt jun., and M. Bicknell calculated \(\sum_{n=0}^m F_{2^n}^{-1}\) and \(\sum_{n=0}^m F_{k2^n}^{-1}\) respectively. For the Lucas sequences \(u_{n+2}= Au_{n+1}- Bu_n\), \(v_{n+2}= Av_{n+1}- Bv_n\) with initial terms \(u_0=0\), \(u_1=1\), \(v_0=2\), \(v_1=A\), W. E. Greig calculated \(\sum_{n=0}^m u_{k2^n}^{-1}\) in the case \(B=-1\) and R. S. Melham and A. G. Shannon in the case \(B=1\). R. André-Jeanin calculated \(\sum_{n=1}^\infty 1/(u_{kn} u_{k(n+1)})\) and \(\sum_{n=1}^\infty 1/(v_{kn} v_{k(n+1)})\) in the case \(B=-1\) and \(2\nmid k\). Melham and Shannon calculated the same sums in the case \(B=1\). NEWLINENEWLINENEWLINEIn this paper the authors consider a general second-order recurrent sequence \(w_{n+2}= Aw_{n+1}- Bw_n\) and calculate its reciprocal sums \(\sum_{n=0}^{m-1} \frac{B^{f(n)}u_{\Delta f(n)}} {w_{f(n)} w_{f(n+1)}}\), where \(f\) is an arithmetic function and \(\Delta f(n)= f(n+1)- f(n)\). The results imply all of the above.