Representations with k-generalized Fibonacci numbers

Abstract

We study representations of integers using $k$-generalized Fibonacci numbers. For $k\ge 2$, we first consider signed representations of zero with coefficients in $\{-1,0,1\}$ and give a recursive description of their number. The resulting counting sequences satisfy linear recurrences whose characteristic polynomials are determined explicitly. In the Fibonacci and Tribonacci cases, these recurrences reveal unexpected connections between the corresponding representation counts and Tribonacci and Fibonacci sequences, respectively. We then consider representations with coefficients in $\{0,1\}$ in the Tribonacci case. Using a random inhomogeneous Tribonacci recurrence, we construct a binary-tree model in which representation multiplicities are encoded by a family of polynomials satisfying the product formula $Q_n(x)={\prod }_{k=2}^{n-1}(1+x^{T_k})$. This product admits a probabilistic interpretation in terms of weighted Bernoulli sums. After normalization by $T_n$, these sums converge in distribution to the Bernoulli convolution ${\sum }_{j=1}^{\infty }{\epsilon }_j{\rho }^{-j}$, where $\rho$ is the Tribonacci constant and the ${\epsilon }_j$ are independent Bernoulli random variables. The limiting distribution satisfies a natural self-similarity relation, linking the representation problem to self-similar measures associated with Tribonacci scaling.

20 pages, 4 figures