On Some identities for Generalized Fibonacci and Lucas Sequences with Rational Subscript (Journal of Informatics and Mathematical Sciences, 2019)
In this paper, we exploit general techniques from matrix theory to establish some identities for generalized Fibonacci and Lucas sequences with rationalsubscripts of the forms \(\frac{n}{2}\) and \(\frac{r}{s}\). For this purpose, we consider matrix functions \(X\mapsto X^{n/2}\) (resp. \(X\mapsto X^{r/s}\)) of two special matrices, and discuss whether the \(\frac{n}{2}\) (resp. \(\frac{r}{s}\)) are integers or irreducible fractions.
Keywords: Horadam Sequences; Generalized Fibonacci Sequences; Generalized Lucas Sequences; Matrix Functions