Fibonacci series with several parameters (Q1728620)

The authors introduce a sequence $\{F_{x,n}\}_{n\ge 0}$ which is linearly recurrent of order $m$, where $x=(x_1,\dots, x_m)$ is a vector encoding the coefficients of the recurrence. The authors find various examples of such sequences expressed in terms of solutions to difference equations and deduce several summation identities for members of such sequences. The proofs are based on straightforward algebraic manipulations. Some identities with generalised $m$-Fibonacci numbers (in which each term is the sum of the previous $m$ terms) are derived as a byproduct. The paper contains several numerical examples. For the entire collection see [Zbl 1398.35005].