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Fibonacci and Lucas numbers as cumulative connection constants. (English) Zbl 0979.11009

Given two sequences \(\{r_n\}\) and \(\{s_n\}\) of complex numbers (where \(n\geq 1\)), one may generate two associated polynomial sequences: \(\{q_n(x)\}\), \(\{p_n(x)\}\), where \(q_0(x)= p_0(x) =1\) and \[ q_n(x) = (x - r_n)q_{n-1}(x),\quad p_n(x) = (x - s_n)p_{n-1}(x),\quad n\geq 1. \] In addition, let “connection constants” \(L_{n,k}\) be defined by \(p_n(x)=\sum^n_{k=0} L_{n,k}q_n(x).\) Let the \(n^{\text{th}}\) cumulative connection constant \(C_n\) be defined by \(C_n=\sum^n_{k=0}L_{n,k}\). In separate examples, the authors find sequences \(\{r_n\}\) and \(\{s_n\}\) such that \(C_n = F_n\), \(L_n\) (the \(n^{\text{th}}\) Fibonacci and Lucas numbers, respectively).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B37 Recurrences
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