基于k阶广义斐波那契数的表示
Representations with k-generalized Fibonacci numbers
浏览论文内容
中文总结 AI 辅助
该研究探讨整数的k阶广义斐波那契数表示，给出系数为{-1,0,1}的零表示计数的递推关系，构造系数为{0,1}的 tribonacci 表示的二叉树模型，揭示其与自相似测度的联系。
中文摘要 AI 辅助
我们研究整数的k阶广义斐波那契数表示。对于k≥2，首先考虑系数在{-1,0,1}中的零的符号表示，并给出其数量的递归描述，所得计数序列满足线性递推关系，其特征多项式被明确确定。在斐波那契和 tribonacci 情形中，这些递推关系分别揭示了相应表示计数与 tribonacci 和斐波那契序列之间的意外联系。接着考虑 tribonacci 情形下系数在{0,1}中的表示，利用随机非齐次 tribonacci 递推关系，我们构造了一个二叉树模型，其中表示重数由满足乘积公式Qₙ(x)=∏ₖ=2ⁿ⁻¹(1+x^{Tₖ})的多项式族编码。该乘积可根据加权伯努利和给出概率解释，经Tₙ归一化后，这些和依分布收敛到伯努利卷积∑ⱼ=1^∞εⱼρ⁻ʲ，其中ρ为 tribonacci 常数，εⱼ为独立伯努利随机变量。极限分布满足自然的自相似关系，将该表示问题与 tribonacci 缩放相关联的自相似测度联系起来。
英文摘要
We study representations of integers using $k$-generalized Fibonacci numbers. For $k\geq 2$, 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 $\sum_{j=1}^{\infty}\varepsilon_jρ^{-j}$, where $ρ$ is the Tribonacci constant and the $\varepsilon_j$ 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.
