$k$-Lucas Annulus for Polynomial Zeros

Abstract

We prove a binomial identity relating the $k$-Lucas and $k$-Fibonacci sequences: for real $k>0$ and integers $m\ge 1$, $n\ge 1$, a binomial-weighted sum of $k$-Lucas numbers reduces to a multiple of $L_{k,mn}$ when $n$ is even, but to a multiple of $F_{k,mn}$ when $n$ is odd. This parity dependence traces back to the $k$-Lucas Binet formula lacking the normalising factor $1/(\alpha -\beta )$ present for $k$-Fibonacci numbers. The identity supplies, for each parity of $n$ and each $m$, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a $k$-Lucas annulus containing all the zeros of a complex polynomial of degree $n$, complementing the $k$-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.

arXiv commentThe k-Lucas sequence is a particular Horadam sequence, and the identity in Theorem 1.1 follows as a specialization of known binomial identities for Horadam sequences, which I had not seen at first