Codenominator function

Codenominator Function and the involution Jimm

The codenominator is a function that generalizes the Fibonacci sequence to the index set of positive rational numbers, $\mathbb {Q} ^{+}$. Many known Fibonacci identities carries over to the codenominator. It is related to Dyer's outer automorphism of $PGL(2,\mathbb {Z} )$. One can express the equivariant real modular form Jimm in terms of the codenominator.

Definition

The codenominator function$F:\mathbb {Q} ^{+}\to \mathbb {Z} ^{+}$ is defined by the following system of functional equations:

1. $F\left(1+{\frac {1}{x}}\right)=F(x)$

2. $F(\left({\frac {1}{1+x}}\right)=F(x)+F\left({\frac {1}{x}}\right)$,

with the initial condition $F(1)=1$. (The name `codenominator' comes from the fact that the usual denominator function $F:\mathbb {Q} ^{+}\to \mathbb {Z} ^{+}$ is defined by the functional equations $D(1+x)=D(x)$, $D(1/(1+x))=D(x)+D(1/x)$, and the initial condition $D(1)=1$.)

For integers $n$, the codenominator agrees with the standard Fibonacci sequence, satisfying the recurrence relation:

$F_{n+1}=F_{n}+F_{n-1},$

where $F_{1}=F_{2}=1$. The codenominator function extends this sequence to positive rational inputs using continued fractions.

Properties

1. Fibonacci recursion: Codenominator satisfies the Fibonacci recurrence for rational inputs:

 $F(x+n)=F_{n}F(x+1)+F_{n-1}F(x),$

where $n>1$ is an integer, $x\in \mathbb {Q} ^{+}$ and $F_{n}$ is the $n$th Fibonacci number.

2. Fibonacci invariance: For any integer $n>1$ and $x\in \mathbb {Q} ^{+}$

$F\left({\frac {F_{n}+F_{n+1}x}{F_{n-1}+F_{n}x}}\right)=F(x).$

3. Symmetry: If $x\in (0,1)\cap \mathbb {Q}$, then $F(1-x)=F(x).$

4. Continued Fractions: For a rational number $x$ expressed as a continued fraction $[n_{0},n_{1},...,n_{k}]$, the value of $F(x)$ can be computed recursively using Fibonacci numbers as:

$F[n_{0},n_{1},\ldots ,n_{k}]=F_{n_{0}}F[n_{1},\ldots ,n_{k}]+F_{n_{0}-1}F[n_{1}+1,\ldots ,n_{k}].$

5. Periodicity: For any positive integer $m$, the codenominator $F[n_{0},n_{1},\ldots ,n_{k}]$ is periodic modulo at most $\pi (m)$ in each variable $n_{k}$, where $\pi (m)$ is the Pisano period.

Relation to Dyer's outer automorphism

The Jimm (ج) function is defined on positive rational arguments via

$J(x):={\frac {F(1/x)}{F(x)}}.$

It admits an extension to the set of non-zero real numbers. This extension is continuous at irrationals, has jumps at rationals, is differentiable a.e. and with derivative vanishing a.e. Moreover this extension satisfies the functional equations

1. Involutivity $J(J(x))=J(x)$ (except on the set of golden irrationals)

2. Covariance with $1-x$: $J(1-x)=1-J(x)$

3. Covariance with $1/x$: $J(1/x)=1/J(x)$

4. Covariance with $-x$: $J(-x)=-1/J(x)$

Since the extended modular group $PGL(2,\mathbb {Z} )$ is generated by the involutions $1-x$, $1/x$ and $-x$, Jimm is a `real' covariant modular form. In fact Jimm is a representation of Dyer's outer automorphism of $PGL(2,\mathbb {Z} )$.

Applications

Jimm sends real quadratic irrationals to real quadratic irrationals, except the noble irrationals, which it sends to rationals in a 2-1 manner. It commutes with the Galois conjugation on this set. Jimm conjecturally sends algebraic numbers of degree $>2$ to transcendental numbers.

See Also

• Fibonacci sequence
• Continued fraction
• Modular form
• Farey sequence
• Pisano period
• Golden numbers
• Quadratic irrationals

References

• Uludağ, A. M., & Gökmen, B. E. (2022). "The Conumerator and the Codenominator." Bulletin des Sciences Mathématiques, 180, Article 103192. DOI: [10.1016/j.bulsci.2022.103192](https://doi.org/10.1016/j.bulsci.2022.103192).