Draft:Codenominator function

• Comment: This draft has only one reference. More than one reference is needed.
  A review is being requested at WikiProject Mathematics. Robert McClenon (talk) 01:52, 1 December 2024 (UTC)

• Comment: The Isola reference, as well as predating the supposed introduction of this concept, is in a predatory journal and cannot be used. —David Eppstein (talk) 21:54, 1 December 2024 (UTC)

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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 of the codenominator

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

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

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

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

$$D(1+x)=D(x),\quad D(1/(1+x))=D(x)+D(1/x),$$

and the initial condition $D(1)=1$.)

Connection with the Fibonacci sequence 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 extends this sequence to positive rational inputs using continued fractions. Moreover, for every rational $x\in \mathbb {Q} ^{+}$, the sequence $G_{n}:=F(x+n)$ is the so-called `Gibonacci' sequence defined by $G_{0}:=F(x)$, $G_{1}:=F(1/x)$ and the recursion $G_{n+2}=G_{n}+G_{n-1}$.

Examples

1. $F(n)=F_{n}$ for integral $n>0$.

2. $F(1/2)=2$, more generally $F(1/n)=F(n+1)=F_{n+1}$ for integral $n>0$.

3. $F(n+1/2)=L_{n}=F_{n+1}+F_{n-1}$, where $L_{n}$ is the Lucas sequence OEIS: A000204.

4. $F(n+1/3)=2F_{n+1}+F_{n-1}$ is the sequence OEIS: OEIS:A001060.

5. $F(n+1/4)=3F_{n+1}+F_{n-1}$ is the sequence OEIS: OEIS:OEIS:A022121.

6. $F(n+1/5)=5F_{n+1}+F_{n-1}$ is the sequence OEIS: OEIS:OEIS:A022138.

7. $F(n+1/n)=2F_{n}^{2}+(-1)^{n}$ is the sequence OEIS: OEIS:OEIS:A061646.

8. $F(23/31)=107$, $F(31/23)=47$.

9. $F(19/41)=2^{4}\times 7$, $F(41/19)=3^{4}$.

10. $F({\frac {F_{n}}{F_{n+1}}})=n,\quad$ $F({\frac {F_{n+1}}{F_{n}}})=1$.

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.

2-variable form of functional equations: The functional equations (1-4) can be written in the two-variable form as follows:

$J(x)=y\iff J(y)=x$

$xy=1\iff J(x)J(y)=1$

$x+y=0\iff J(x)J(y)=-1$

$x+y=1\iff J(x)+J(y)=1$

The following equation is also valid:

 ${\frac {1}{x}}+{\frac {1}{y}}=1\iff {\frac {1}{J(x)}}+{\frac {1}{J(y)}}=1$

Involution Jimm and relation to Dyer's outer automorphism

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

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

The function J 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. Jimm conjugates the Gauss map $G$ (see Gauss–Kuzmin–Wirsing operator) to the so-called Fibonacci map $\Phi$ , i.e. $\Phi =J\circ G\circ J$.

Moreover this extension satisfies the functional equations

1. Involutivity

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

2. Covariance with $1-x$:

$$J(1-x)=1-J(x)$$ (provided $x\neq 1$)

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 $-x$, Equations (1-4) express the fact that Jimm is a `real' modular covariant form. In fact Jimm is a representation of Dyer's outer automorphism of $PGL(2,\mathbb {Z} )$.

Properties of the plot of Jimm The plot of Jimm hides many copies of the golden ratio $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ in it. For example

1. $\lim _{x\to +\infty }J(x)=\varphi$, $J(\varphi )=+\infty$

2. $\lim _{x\to -\infty }J(x)={\bar {\varphi }}$, $J({\bar {\varphi }})=-\infty$

3. $\lim _{x\to 0+}J(x)=\varphi ^{-1}$, $J(\varphi ^{-1})=0$

4. $\lim _{x\to 0-}J(x)={\bar {\varphi }}^{-1}$, $J({\bar {\varphi }}^{-1})=0$

5. $\lim _{x\to 1+}J(x)=1+\varphi$, $J(1+\varphi )=1$

6. $\lim _{x\to 1-}J(x)=1+{\bar {\varphi }}$, $J(1+{\bar {\varphi }})=1$

In general, for any rational $q$, the limit $\lim _{x\to q+}J(x)$ will be of the form ${\frac {a\varphi +b}{c\varphi +d}}$ with $a,b,c,d\in \mathbb {Z}$ and $ad-bc=\pm 1$. The limit $\lim _{x\to q-}J(x)$ will be its Galois conjugate ${\frac {a{\bar {\varphi }}+b}{c{\bar {\varphi }}+d}}$.

Jimm on real quadratic irrational numbers

Jimm sends real quadratic irrationals to real quadratic irrationals, except the golden irrationals, which it sends to rationals in a 2–1 manner. It commutes with the Galois conjugation on the set of non-golden quadratic irrationals. If $x=a+{\sqrt {b}}\quad (a,b\in \mathbb {Q} )$ is a real quadratic irrational, which is not a golden number, then

1. $N(x)=1\iff N(J(x))=1$

2. $Tr(x)=0\iff N(J(x))=-1$

3. $Tr(x)=1\iff Tr(J(x))=1$

4. $Tr({1}/{x})=1\iff Tr({1}/{J(x)})=1$

where $N(x):=a^{2}-b$ denotes the norm and $Tr(x):=2a$ denotes the trace of $x$.

Jimm on higher algebraic numbers: Jimm conjecturally sends algebraic numbers of degree $>2$ to transcendental numbers.

See also

• Fibonacci sequence
• Continued fraction
• Modular form
• Farey sequence
• Pisano period
• Golden number
• Quadratic irrational

References

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Categories:

• Number theory
• Mathematical functions
• Fibonacci sequence