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 extends 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 $\alpha$ of the extended modular group PGL(2, Z). One can express the real $\alpha$-covariant modular function Jimm (defined below) in terms of the codenominator. Jimm induces an involution of the moduli space of rank-2 pseudolattices and is related to the arithmetic of real quadratic irrationals.

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

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

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

The codenominator takes every positive integral value infinitely often.

Connection with the Fibonacci sequence

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

$$F_{n+1}=F_{n}+F_{n-1},\quad F_{1}=F_{2}=1.$$

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

	Examples ($n$ is a positive integer)
1	$F(n)=F_{n}$
2	$F(1/2)=2$, more generally $F(1/n)=F(n+1)=F_{n+1}$.
3	$F(n+1/2)=L_{n}=F_{n+1}+F_{n-1}$ is the Lucas sequence OEIS: A000204.
4	$F(n+1/3)=2F_{n+1}+F_{n-1}$ is the sequence OEIS: A001060.
5	$F(n+1/4)=3F_{n+1}+F_{n-1}$ is the sequence OEIS: A022121.
6	$F(n+1/5)=5F_{n+1}+F_{n-1}$ is the sequence OEIS: A022138.
7	$F(n+1/n)=2F_{n}^{2}+(-1)^{n}$ is the sequence 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:

$$F(x+n)=F_{n}F(x+1)+F_{n-1}F(x),\quad (n>1,\quad x\in \mathbb {Q} ^{+})$$

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. Reversion:

  $$F[0,n_{1},\ldots ,n_{k}]=F[0,n_{k},\ldots ,n_{1}]$$

6. Periodicity: For any positive integer $m$, the codenominator $F[n_{0},n_{1},\ldots ,n_{k}]$ is periodic in each partial quotient $n_{k}$ modulo $m$ with period divisible with $\pi (m)$ , where $\pi (m)$ is the Pisano period.

7. Fibonacci identities: Many known Fibonacci identities admit a codenominator version. For example, if at least two among $x,y,z\in \mathbb {Q} ^{+}$ are integral, then

$$F({x+y})F({x+z})-F(x)F({x+y+z})=\mathrm {cds} (x)F({y})F(z),$$

where $\mathrm {cds} (x):=F(1/x)^{2}-F(1/x)F(x)-F(x)^{2}$ is the codiscriminant (called 'characteristic number' in ). This reduces to Tagiuri's identity when $x,y,z\in \mathbb {Z}$; which in turn is a generalization of the famous Catalan identity. Any Gibonacci identity can be interpreted as a codenominator identity. There is also a combinatorial interpretation of the codenominator.

Involution Jimm

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

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

This function is involutive and admits a natural extension to non-zero rationals via $J(-x):=-1/J(x)$ which is also involutive.

Let $x=[n_{0},n_{1},\dots ,n_{k}]>1$ be the simple continued fraction expansion of $x\in \mathbb {Q} ^{+}$. Denote by $1_{k}$ the sequence $1,1,\dots ,1$ of length $k$. Then:

 $J(x)=[1_{n_{0}-1},2,1_{n_{1}-2},2,1_{n_{2}-2},2,\dots ,2,1_{n_{k-1}-2},2,1_{n_{k}-1}]$

with the rules:

$[\dots ,n,1_{0},m,\dots ]:=[\dots ,n,m,\dots ]$

and

$[\dots ,n,1_{-1},m,\dots ]:=[\dots ,n+m-1,\dots ]$.

The function $J$ admits an extension to the set of non-zero real numbers by taking limits (for positive real numbers one can use the same rules as above to compute it). This extension (denoted again $J$) is 2-1 valued on golden -or noble- numbers (i.e. the numbers in the PGL(2, Z)-orbit of the golden ratio $\varphi ={\frac {1+{\sqrt {5}}}{2}}$).

Moreover, this extension

sends rationals to rationals,

sends golden numbers to rationals,

is involutive except on the set of golden numbers,

sends real quadratic irrationals (except golden numbers) to real quadratic irrationals (see below),

commutes with the Galois conjugation on real quadratic irrationals (see below),

is continuous at irrationals,

has jumps at rationals,

is differentiable a.e.,

has vanishing derivative a.e.,

sends a set of full measure to a set of null measure and vice versa

and moreover satisfies the functional equations

Involutivity

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

Covariance with $1-x$

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

Covariance with $1/x$

$J(1/x)=1/J(x)$,

Covariance with $-x$

$J(-x)=-1/J(x)$.

These four functional equations in fact characterize Jimm.

Dyer's outer automorphism and Jimm

The extended modular group PGL(2, Z) admits the presentation

 $\langle U,V,K\,|\,U^{2}=V^{2}=K^{2}=(UV)^{2}=(KU)^{3}=1\rangle$

where (viewing PGL(2, Z) as a group of Möbius transformations) $U:=x\to 1/x$, $V:x\to -x$ and $K:x\to 1-x$.

The map $\alpha$ of generators

 $\alpha :U\to U,\quad V\to UV,\quad K\to K$ 

defines an involutive automorphism PGL(2, Z) $\to$ PGL(2, Z), called Dyer's outer automorphism. It is known that Out(PGL(2, Z))$\simeq \mathbf {Z} /2\mathbf {Z}$ is generated by $\alpha$. The modular group PSL(2, Z) $<$ PGL(2, Z) is not invariant under $\alpha$.

Dyer's outer automorphism can be expressed in terms of the codenumerator, as follows: Suppose $p,q,r,s\in \mathbf {Z} ^{+}$ and $M={\begin{bmatrix}p&q\\r&s\end{bmatrix}}\in \mathrm {PGL} _{2}(\mathbf {Z} )$. Then

 $\alpha (M)={\begin{bmatrix}\mathrm {F} \left({\frac {2r+s}{2p+q}}\right)-\mathrm {F} \left({\frac {r+s}{p+q}}\right)&2\mathrm {F} \left({\frac {r+s}{p+q}}\right)-\mathrm {F} \left({\frac {2r+s}{2p+q}}\right)\\\mathrm {F} \left({\frac {2p+q}{2r+s}}\right)-\mathrm {F} \left({\frac {p+q}{r+s}}\right)&2\mathrm {F} \left({\frac {p+q}{r+s}}\right)-\mathrm {F} \left({\frac {2p+q}{2r+s}}\right)\end{bmatrix}}.$

The covariance equations above implies that $J$ is a representation of $\alpha$ as a map P(R) $\to$ P(R), i.e. $J(Mx)=\alpha (M)J(x)$ whenever $x\in \mathbb {R}$ and $M\in$PGL(2, Z). Another way of saying this is that $J$ is a $\alpha$-covariant map.

In particular, $J$ sends PGL(2, Z)-orbits to PGL(2, Z)-orbits, thereby inducing an involution of the moduli space of rank-2 pseudo lattices , PGL(2, Z)\ P(R), where P(R) is the projective line over the real numbers.

Given $x,y,\in$ P(R), the involution $J$ sends the geodesic on the hyperbolic upper half plane ${\mathcal {H}}$ through $x,y$ to the geodesic through $J(x),J(y)$, thereby inducing an involution of geodesics on the modular curve PGL(2, Z)\${\mathcal {H}}$. It preserves the set of closed geodesics because $J$ sends real quadratic irrationals to real quadratic irrationals (with the exception of golden numbers, see below) respecting the Galois conjugation on them.

2-variable form of functional equations: The functional equations can be written in the two-variable form as :

Involutivitiy

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

Covariance with $1-x$

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

Covariance with $1/x$

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

Covariance with $-x$

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

As a consequence of these, one has: ${\frac {1}{x}}+{\frac {1}{y}}=1\iff {\frac {1}{J(x)}}+{\frac {1}{J(y)}}=1.$

Properties of the plot of Jimm and the golden ratio:

By involutivity, the plot of $J$ is symmetric with respect to the diagonal $x=y$, and by covariance with $1-x$, the plot is symmetric with respect to the diagonal $x+y=1$. The fact that the derivative of $J$ is 0 a.e. can be observed from the plot.

The plot of Jimm hides many copies of the golden ratio $\varphi$ 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	$\lim _{x\to 1^{-}}J(x)=1+{\bar {\varphi }},$	$J(1+{\bar {\varphi }})=1$

More generally, for any rational $q$, the limit $\lim _{x\to q+}J(x)$ is of the form $A:={\frac {a\varphi +b}{c\varphi +d}}$ with $a,b,c,d\in \mathbb {Z}$ and $ad-bc=\pm 1$. The limit $\lim _{x\to q-}J(x)$ is its Galois conjugate ${\bar {A}}:={\frac {a{\bar {\varphi }}+b}{c{\bar {\varphi }}+d}}$. Conversely, one has $J(A)=J({\bar {A}})=q$.

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, i.e. if $J(a+{\sqrt {b}})=A+{\sqrt {B}}$, then $J(a-{\sqrt {b}})=A-{\sqrt {B}}$, with $a,b,A,B\in \mathbb {Q}$ and $b,B$ positive non-squares.

This is a highly non-trivial map, for example

 $J\left({\frac {3+5{\sqrt {2}}}{7}}\right)={\frac {-3+2{\sqrt {95}}}{7}},\quad J({\sqrt {11}})={\frac {15+{\sqrt {901}}}{26}}.$

If $x=a+{\sqrt {b}}\quad (a,b\in \mathbb {Q} )$ is a real quadratic irrational, which is not a golden number, then as a consequence of the functional equations of $J$ one has

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 $x$.

On the other hand, $J$ may send two members of one real quadratic number field to members of two different real quadratic number fields; i.e. it does not respect individual class groups.

Jimm on Markov irrationals

Jimm sends the Markov irrationals to 'simpler' quadratic irrationals, see table below.

Markov number	Markov irrational $x$	$J(x)$
1	$(1+{\sqrt {5}})/2$	$\infty$
2	$1+{\sqrt {2}}$	${\sqrt {2}}$
5	$(9+{\sqrt {221}})/10$	${\sqrt {6}}-1$
13	$(23+{\sqrt {1517}})/26$	${\sqrt {12}}-2$
29	$(53+{\sqrt {7565}})/58$	${\sqrt {35/6}}-1$
34	$(15+5{\sqrt {26}})/17$	${\sqrt {20}}-3$
89	$(157+{\sqrt {71285}})/178$	${\sqrt {30}}-4$
169	$(309+{\sqrt {257045}})/338$	${\sqrt {204/35}}-1$
194	$(86+{\sqrt {21170}})/97$	${\sqrt {119/10}}-2$
233	$(411+{\sqrt {488597}})/466$	${\sqrt {42}}-5$
433	$(791+{\sqrt {1687397}})/866$	$(12{\sqrt {143}}-60)/59$
610	$(269+{\sqrt {209306}})/305$	${\sqrt {56}}-6$
985	$(1801+{\sqrt {8732021}})/1970$	${\sqrt {1189/204}}-1$
1325	$(2339+{\sqrt {15800621}})/2650$	$(3{\sqrt {434}}-42)/14$
1597	$(2817+{\sqrt {22953677}})/3194$	${\sqrt {72}}-7$
2897	$(5137+{\sqrt {75533477}})/5794$	${\sqrt {12958}}/33-2$
4181	$(7375+{\sqrt {157326845}})/8362$	${\sqrt {90}}-8$
5741	$(10497+5{\sqrt {11865269}})/11482$	$3{\sqrt {915530}}/1189-1$
6466	$(2953+5{\sqrt {940706}})/3233$	$(7{\sqrt {10295}}-297)/292$
7561	$(13407+{\sqrt {514518485}})/15122$	$(4{\sqrt {14430}}-279)/139$
9077	$(16013+{\sqrt {741527357}})/18154$	$7{\sqrt {22}}/6-4$
10946	$(19308+4{\sqrt {67395890}})/21892$	${\sqrt {110}}-9$
14701	$(26879+{\sqrt {1945074605}})/29402$	$(70{\sqrt {4899}}-2030)/2029$
28657	$(50549+{\sqrt {7391012837}})/57314$	${\sqrt {132}}-10$
33461	$(61181+{\sqrt {10076746685}})/66922$	$(13{\sqrt {184030}}/2310-1)$
37666	$(17202+5{\sqrt {31921370}})/18833$	$(8{\sqrt {1081730}}-3479)/3421$
43261	$(76711+{\sqrt {16843627085}})/86522$	$13{\sqrt {345}}/70-2$

Jimm and dynamics

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

The expression of Jimm in terms of continued fractions shows that, if a real number $x$ obeys the Gauss-Kuzmin distribution, then the proportion of 1's among the partial quotients of $J(x)$ is one, i.e. $J(x)$ does not obey the Gauss-Kuzmin statistics. This argument can be used to show that $J$ sends the set of real numbers obeying the Gauss-Kuzmin statistics, which is of full measure, to a set of null measure.

Jimm on higher algebraic numbers

It is widely believed that if $x$ is an algebraic number of degree $>2$, then it obeys the Gauss-Kuzmin statistics. By the above remark, this implies that $J(x)$ violates the Gauss-Kuzmin statistics. Hence, according to the same belief, $J(x)$ must be transcendental. This is the basis of the conjecture that Jimm sends algebraic numbers of degree $>2$ to transcendental numbers. A stronger version of the conjecture states that any two algebraically related $J(x)$, $J(y)$ are in the same PGL(2, Z)-orbit, if $x,y$ are both algebraic of degree $>2$.

Functional equations and equivariant modular forms

Given a representation $\rho :\mathrm {PSL} _{2}(\mathbf {Z} )\to \mathrm {PGL} _{2}(\mathbf {Z} )$, a meromorphic function $h$ on ${\mathcal {H}}$ is called a $\rho$-covariant function if

$h(\gamma \cdot z)=\rho (\gamma )\cdot h(z)\quad {\text{for all }}z\in \mathbb {H} ,\,\gamma \in \Gamma .$

(sometimes $h$ is also called a $\rho$-equivariant function) It is known that there exists meromorphic covariant functions $h$ on the upper half plane ${\mathcal {H}}$, i.e. functions satisfying $h(Mz)=Mh(z)$. The existence of meromorphic functions satisfying a version of the functional equations for $J$ is also known .

Some codenumerator values

Below is a table of codenominator values $41/k$, where 41 is an arbitrarily chosen number.

$k$	$F(41/k)$
1	$59369\times 2789$
2	$7\times 2161$
3	$5\times 7\times 19$
4	$5\times 67$
5	$11\times 19$
6	$3\times 5\times 7$
7	$103$
8	$3^{2}\times 23$
9	$31$
10	$7^{3}$
11	$17$
12	$2^{3}\times 3$
13	$5\times 13$
14	$3\times 281$
15	$23$
16	$19$
17	$29$
18	$17$
19	$3^{4}$
20	$89\times 199$
21	$3\times 5\times 11\times 41$
22	$31$
23	$2\times 3$
24	$2^{2}\times 3$
25	$2^{2}$
26	$7$
27	$233$
28	$47$
29	$17$
30	$13$
31	$199$
32	$17$
33	$131$
34	$2\times 3\times 11$
35	$2^{6}$
36	$3\times 43$
37	$3^{2}\times 23$
38	$3\times 137$
39	$9349$
40	$3\times 5\times 7\times 11\times 41\times 2161$
41	$1$

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