Approximation of \(\infty\)-generalized Fibonacci sequences and their asymptotic Binet formula (Q2719880)

Let \(\{a_j\}_{j\geq 0}\) and \(\{\alpha_j\}_{j\geq 0}\) be sequences of real or complex numbers, with not all \(\alpha_j=0\). An \(r\)-generalized Fibonacci sequence is defined by: NEWLINE\[NEWLINEV_n^{(r)}= \begin{cases} \sum_{j=0}^r a_j V_{n-j-1} &\text{if }n\geq 1,\\ \alpha_n &\text{if }-r< n\leq 0, \end{cases}NEWLINE\]NEWLINE whereas an infinity-generalized Fibonacci sequence is defined by: NEWLINE\[NEWLINEV_n= \begin{cases} \sum_{j=0}^\infty a_j V_{n-j-1} &\text{if }n\geq 1,\\ \alpha_n &\text{if }n\leq 0. \end{cases}NEWLINE\]NEWLINE Among other results, the authors show that for all \(n\geq 1\), \(V_n\) exists if and only if \(V_n= \lim_{r\to\infty} V_n^{(r)}\).