Identically vanishing $k$-generalized Fibonacci polynomials

TL;DR

This work analyzes the downward extension of the $k$-generalized Fibonacci polynomial recurrence and shows that identically vanishing polynomials occur at a complete finite set of negative indices, amounting to $k(k-1)/2$ cases. It introduces a left-justified $k$-nomial triangle for $n<0$, derives concise multinomial sums for positive and negative indices, and provides explicit expressions for polynomial coefficients via elementary symmetric polynomials of the roots. A factorization framework $\mathcal{F}_{n,k}(x)=x^{r_{n,k}}(x^k+1)^{\rho_{n,k}}Q_{n,k}(x^k)$ and detailed root analyses reveal rotational symmetry, monotonicity properties of degree, and real-root behavior dependent on parity of $k$ and the sign of $n$. The paper also connects these polynomials to the $k$-generalized Jacobsthal and Pell families, offering generating-function based sums and practical computational forms. Overall, it extends the theory of generalized Fibonacci polynomials to negative indices with rigorous characterization, structural results, and computationally efficient representations.

Abstract

The recurrence for the $k$-generalized Fibonacci polynomials is usually iterated upwards to positive values of $n$ only. When the recurrence is iterated downwards to $n<0$, there are indices where the polynomials vanish identically. This fact does not seem to have been noted in the literature. We derive the set of such indices. We present the left-justified generalized Pascal triangle for $n<0$. For $k\ge3$ and $n<0$, we show that the degree of the polynomial does not increase monotonically with $|n|$. We derive expressions for the individual polynomial coefficients (the elementary symmetric polynomials of the roots). We present results for the properties of the polynomials, for both $n>0$ and $n<0$, including factorization of the polynomials and properties of the roots (including bounds on the amplitudes of the nonzero roots). Results are also derived for real roots. (Separate treatments are required for $n>0$ and $n<0$.) We employ generating functions to derive new combinatorial sums for the polynomials. The sums are more concise and computationally more efficient than previously published expressions. We also exhibit the relation of the $k$-generalized Jacobsthal and Pell polynomials to the Fibonacci polynomials.

Figures (3)

• Figure 1: Plot of the degree $d_{n,k}$ of a polynomial for $k=5$ and $n>0$ (dotdash line) and $n<0$ (solid line and circles). The value of $|n|$ is plotted on the horizontal axis. The blanks for $n<0$ occur when a polynomial vanishes identically.
• Figure 2: Argand diagram plots of the roots of the polynomials $\mathcal{F}_{-40,3}(x)$ and $\mathcal{F}_{40,8}(x)$.
• Figure 3: Plot of $\zeta_{n,k}$ for (i) $k=2,3,8$ and $n>0$, and (ii) $k=4$ and $n<0$.

Theorems & Definitions (36)

• Theorem 1.1
• proof
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Proposition 3.1
• Remark 3.2
• Proposition 3.3