Inverses of Fibonacci and Lucas Numbers via Rational Indices

Technical Summary: Inverses of Fibonacci and Lucas Numbers via Rational Indices

Problem Statement
The paper addresses the extension of classical Fibonacci ($F_n$) and Lucas ($L_n$) sequences to rational indices $X \in \mathbb{Q}_{>0}$. While previous work by Uludağ and Gökmen (2022) established that rational-indexed Fibonacci numbers ($F_X$) can be expressed using the "codenominator" function $F$, and that infinitely many such representations exist, the specific structural properties, explicit formulas for arbitrary rational indices, and the existence of multiplicative inverses within this framework remained to be fully characterized. The authors seek to derive general explicit formulas for $F_X$, determine conditions under which $F_X$ coincides with classical Lucas numbers or deviates from the integer-indexed Fibonacci sequence, and prove the existence of multiplicative inverses for these numbers.

Methodology
The study relies on the codenominator function $F: \mathbb{Q}_{>0} \to \mathbb{Z}_{>0}$, defined recursively via the conumerator function. The function $F$ is constructed using properties of continued fractions. For a rational number $X = [n_0; n_1, \dots, n_k]$, the authors utilize the recursive relations of $F$ (specifically $F(1+X) = F(1/X)$ and $F(\frac{1}{1+X}) = F(X) + F(1/X)$) to decompose rational indices into integer components.

The methodology proceeds in three stages:

1. Derivation of Explicit Formulas: The authors derive closed-form expressions for $F_X$ where the continued fraction expansion of $X$ has lengths up to six. These formulas express $F_X$ as linear combinations of classical Fibonacci ($F_n$) and Lucas ($L_n$) numbers with integer indices determined by the partial quotients of $X$.
2. Analysis of Coincidence and Divergence: By comparing the derived formulas against the definitions of classical sequences, the paper establishes precise conditions under which a rational-indexed value $F_X$ equals a classical Fibonacci number, a Lucas number, or a distinct value.
3. Inverse Computation: The authors investigate the inverses of these rational-indexed numbers. By computing $F_{X^{-1}}$ (where $X^{-1}$ is the reciprocal of the index) and utilizing identities such as Cassini's identity and Binet's formulas, they demonstrate that the product of a rational-indexed number and its inverse yields a unit (specifically $\pm 1$ modulo a related sequence term), thereby establishing the existence of multiplicative inverses.

Key Contributions and Results

• General Explicit Formulas: The paper provides explicit formulas for $F_X$ for continued fractions of length 2 through 6 (Lemmas 3.1–3.4) and a general recursive formula (Lemma 4.2) for arbitrary length $k$. For example, for $X = [n_0; n_1]$, $F_X = F_{n_0}F_{n_1} + F_{n_0-1}F_{n_1+1}$.
• Characterization of Values:
  • Coincidence with Classical Sequences: The paper proves that $F_X$ coincides with a classical Fibonacci number $F_m$ only under specific conditions (e.g., if the continued fraction length $k=1$ and $n_0 \in \{1, 2\}$, or if $k=2$ and $n_0=1$ with specific constraints on $n_1$).
  • Coincidence with Lucas Numbers: It is shown that if $X = [3; n]$, then $F_X = L_{n+1}$. Similarly, $F([2; 1, n]) = L_{n+1}$.
  • Divergence: Corollary 6.1 establishes that for $k \ge 3$, or for $k=1$ with $n_0 \ge 3$ (excluding the Lucas case), $F_X$ is distinct from the classical Fibonacci sequence.
• Existence of Multiplicative Inverses: The central result is the proof that every Fibonacci and Lucas number possesses a multiplicative inverse within the rational-indexed framework.
  • For $F_m = F([1; m])$, the inverse is $F_{m+2}$.
  • For $F_m = F([2; m-2])$, the inverse is $L_{m-1}$.
  • For $L_m = L([3; m-1])$, the inverse is $L_m + F_{m+1}$.
  • The paper demonstrates that $F_X \cdot F_{X^{-1}} \equiv \pm 1 \pmod{\text{related term}}$, confirming the unit property.

Significance
The authors claim that these results extend the rational-indexed viewpoint introduced by Uludağ and Gökmen by providing a systematic method to compute values and inverses for arbitrary rational indices. The work reveals new structural properties of Fibonacci- and Lucas-related sequences, specifically the ability to define and compute multiplicative inverses for these numbers through the codenominator function. The paper suggests that these findings open avenues for further number-theoretic exploration regarding the arithmetic properties of sequences under rational indexation, particularly in the context of modular arithmetic and continued fractions. The authors do not claim immediate applications outside of theoretical number theory but emphasize the enrichment of the algebraic understanding of these classical sequences.