\(r\)-generalized Fibonacci sequences, Cayley-Hamilton theorem and Markov chains (Q2705025)

Let \(a_0, a_1,\dots, a_{r-1}\) be fixed real numbers (some \(r\geq 2\)), and consider the sequence \(\{V_n\), \(n\geq r\}\) defined by the following \(r\)th order linear recurrence relation NEWLINE\[NEWLINEV_{n+1}= a_0V_n+ a_1V_{n-1}+\cdots+ a_{r-1}V_{n-r+1} \quad\text{for }n\geq r-1, \tag \(*\) NEWLINE\]NEWLINE where the numbers \(V_0, V_1,\dots, V_{r-1}\) are specified by initial conditions. Such sequences are called \(r\)-generalized Fibonacci sequences. The aim of this paper is to study the connection between sequences \((*)\) and the properties of suitably defined recurrent Markov chains. On the other hand, using combinatorial arguments, a closed-form expression of \(V_n\) could be derived.