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 $\mathbf {Q} ^{+}$. Many known Fibonacci identities carry over to the codenominator. One can express Dyer's outer automorphism $\alpha$ of the extended modular group $PGL(2,\mathbf {Z} )$ in terms of the codenominator. The real $\alpha$-covariant modular function called Jimm on the real line $\mathbf {R}$ is defined via the codenominator. Jimm induces an automorphism of the Stern–Brocot tree as well as an involution of the moduli space of rank-2 pseudolattices, and it is related to the arithmetic of real quadratic irrationals.

Definition of the codenominator

The codenominator function $F:\mathbf {Q} ^{+}\to \mathbf {Z} ^{+}$ is defined by the following system of functional equations: $$F{\bigl (}1+{\tfrac {1}{x}}{\bigr )}=F(x),\quad F{\bigl (}{\tfrac {1}{1+x}}{\bigr )}=F(x)+F{\bigl (}{\tfrac {1}{x}}{\bigr )}\quad (x\in \mathbf {Q} ^{+})$$ 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 $D:\mathbf {Q} ^{+}\to \mathbf {Z} ^{+}$ can be defined by the functional equations $$D(1+x)=D(x),\quad D{\bigl (}1/(1+x){\bigr )}=D(x)+D{\bigl (}1/x{\bigr )},$$ and the initial condition $D(1)=1$.)

The codenominator takes every positive integer value infinitely often.

Connection with the Fibonacci sequence

For integer arguments, 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 arguments. Moreover, for every rational $x\in \mathbf {Q} ^{+}$, the sequence $G_{n}:=F(x+n)$ is the so-called Gibonacci sequence (also called the generalized Fibonacci sequence) defined by $G_{0}:=F(x),\quad G_{1}:=F(1+x)$ and the recursion $G_{n+2}=G_{n+1}+G_{n}.$

	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 OEIS: A001060.
5	$F(n+1/4)=3F_{n+1}+F_{n-1}$ is OEIS: A022121.
6	$F(n+1/5)=5F_{n+1}+F_{n-1}$ is OEIS: A022138.
7	$F(n+1/n)=2F_{n}^{2}+(-1)^{n}$ is 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	$F{\bigl (}{\tfrac {F_{n}}{F_{n+1}}}{\bigr )}=n,\quad F{\bigl (}{\tfrac {F_{n+1}}{F_{n}}}{\bigr )}=1$.

Properties

1. Fibonacci recursion:

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

1. Fibonacci invariance:

For $n>1$ and $x\in \mathbf {Q} ^{+}$, $$F\!{\Bigl (}{\frac {F_{n}+F_{n+1}x}{F_{n-1}+F_{n}x}}{\Bigr )}\;=\;F(x).$$

1. Symmetry:

If $x\in (0,1)\cap \mathbf {Q}$, then $$F(1-x)=F(x).$$

1. Continued fractions:

For a rational $x$ expressed as a simple continued fraction $[n_{0},n_{1},\dots ,n_{k}]$, the value $F(x)$ can be computed recursively using Fibonacci numbers: $$F[n_{0},n_{1},\ldots ,n_{k}]\;=\;F_{n_{0}}\,F[n_{1},\ldots ,n_{k}]\;+\;F_{n_{0}-1}\,F[n_{1}+1,\ldots ,n_{k}].$$

1. Reversion:

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

1. 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 by $\pi (m)$, where $\pi (m)$ is the Pisano period.

1. Fibonacci identities:

Many known Fibonacci identities admit a codenominator version. For instance, if at least two among $x,y,z\in \mathbf {Q} ^{+}$ 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, a generalization of the famous Catalan identity. Any Gibonacci identity can be interpreted as a codenominator identity. The codiscriminant is a 2-periodic function.

Involution Jimm

The Jimm (ج) function is defined on positive rational inputs by $$J(x)\;:=\;{\frac {F{\bigl (}{\tfrac {1}{x}}{\bigr )}}{F(x)}},\quad x\in \mathbf {Q} ^{+}.$$ This is an involution and naturally extends to nonzero rationals via $J(-x):=-{\tfrac {1}{J(x)}}$, which also remains involutive.

Let $x=[n_{0},n_{1},\dots ,n_{k}]>1$ be a simple continued fraction for $x\in \mathbf {Q} ^{+}$. Denote by $1_{k}$ the sequence of $k$ ones. 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 special rewriting rules $[\dots ,\,n,\,1_{0},\,m,\,\dots ]:=[\dots ,\,n,\,m,\,\dots ],\quad [\dots ,\,n,\,1_{-1},\,m,\,\dots ]:=[\dots ,\,n+m-1,\,\dots ].$

By taking limits, $J$ extends to nonzero reals and remains 2–1 on golden (noble) numbers (i.e., the PGL(2, Z)-orbit of the golden ratio $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$).

From , we have:

• $J$ sends rationals to rationals.
• $J$ sends golden numbers to rationals (2–1).
• $J$ is involutive except on the golden numbers.
• $J$ respects ends of continued fractions: if $x$ and $y$ have the same “tail,” then so do $J(x)$ and $J(y)$.
• $J$ sends real quadratic irrationals to real quadratic irrationals (except the golden ones), commuting with Galois conjugation on them.
• $J$ is continuous at irrationals but has jump discontinuities at rationals.
• $J$ is differentiable almost everywhere, with zero derivative almost everywhere.
• $J$ sends sets of full measure to sets of null measure (and vice versa).

It also satisfies the following functional equations (except for some subtleties at golden numbers):

Involutivity

$J(J(x))=x.$

Covariance with $1-x$

$J(1-x)=1-J(x),$ for $x\neq 1$.

Covariance with $1/x$

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

“Twisted” covariance with $-x$

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

These four properties uniquely characterize Jimm. There is also a “reversion invariance” that says if $J([n_{0},n_{1},\dots ,n_{k}])=[m_{0},m_{1},\dots ,m_{\ell }],$ then $J([n_{k},\dots ,n_{1},n_{0}])=[m_{\ell },\dots ,m_{1},m_{0}].$

Dyer’s outer automorphism and Jimm

The extended modular group $PGL(2,\mathbf {Z} )$ admits the presentation $\langle U,V,K\;|\;U^{2}=V^{2}=K^{2}=(UV)^{2}=(KU)^{3}=1\rangle ,$ where (as Möbius transformations) $U:x\mapsto 1/x$, $V:x\mapsto -x$, and $K:x\mapsto 1-x$.

There is an involutive automorphism $\alpha :PGL(2,\mathbf {Z} )\to PGL(2,\mathbf {Z} )$, called Dyer’s outer automorphism, given on the generators by $\alpha :U\mapsto U,\quad V\mapsto UV,\quad K\mapsto K.$ It is known that $\mathrm {Out} {\bigl (}PGL(2,\mathbf {Z} ){\bigr )}\simeq \mathbf {Z} /2\mathbf {Z}$ is generated by $\alpha$.

Although Failed to parse (unknown function "\lt"): {\displaystyle PSL(2,\mathbf{Z})=\langle VU,\, KU\rangle\lt PGL(2,\mathbf{Z})} is not $\alpha$-invariant, the subgroup $\Gamma =\langle KU,\,UVKV\rangle \subset PSL(2,\mathbf {Z} )$ is. Conjugacy classes of subgroups of $\Gamma$ correspond to bipartite trivalent graphs, and $\alpha$ provides a duality among those graphs.

Dyer’s outer automorphism can be expressed via the codenominator: if $M={\begin{pmatrix}p&q\\r&s\end{pmatrix}}\in PGL(2,\mathbf {Z} ),\quad p,q,r,s\in \mathbf {Z} ^{+},$ then $\alpha (M)={\begin{pmatrix}F({\tfrac {2r+s}{2p+q}})-F({\tfrac {r+s}{p+q}})&2\,F({\tfrac {r+s}{p+q}})\;-\;F({\tfrac {2r+s}{2p+q}})\\F({\tfrac {2p+q}{2r+s}})-F({\tfrac {p+q}{r+s}})&2\,F({\tfrac {p+q}{r+s}})\;-\;F({\tfrac {2p+q}{2r+s}})\end{pmatrix}}.$

From the covariance equations of $J$, it follows $J$ is a representation of $\alpha$ in the sense that $J(Mx)=\alpha (M)\,J(x)$ for $M\in PGL(2,\mathbf {Z} )$ and real $x$. Equivalently, $J$ is an $\alpha$-covariant map.

Hence, $J$ induces an involution on the moduli space of rank-2 pseudo-lattices, PGL(2,\mathbf{Z}) \ P^1(\mathbf{R}), where P^1(\mathbf{R}) is the real projective line.

Geometrically, $J$ sends geodesics in the hyperbolic upper half-plane ${\mathcal {H}}$ to geodesics, so it induces an involution of closed geodesics on the modular curve PGL(2,\mathbf{Z})\setminus \mathcal{H}, since it sends real quadratic irrationals (except golden ones) to real quadratic irrationals, respecting Galois conjugation.

Jimm as a tree automorphism

Djokovic and Miller realized $\mathrm {Aut} {\bigl (}PGL(2,\mathbf {Z} ){\bigr )}$ as a group of automorphisms of the infinite trivalent tree. In that context, $\alpha$ appears as a tree automorphism. Indeed, $\mathrm {Aut} {\bigl (}PGL(2,\mathbf {Z} ){\bigr )}$ is one of seven groups acting with finite vertex stabilizers on that infinite trivalent tree.

Jimm and the Stern–Brocot tree

Bird's tree of rational numbers

Applying Jimm to each node of the Stern–Brocot tree permutes all rationals in a row, while preserving rows, yielding a new tree called the Bird tree.

Reading the denominators of the rationals in the Bird tree level by level gives ‘‘Hinze’s sequence’’: $2,\,3,\,3,\,5,\,4,\,4,\,5,\,8,\,7,\,5,\,7,\,7,\,5,\ldots$ ((sequence 268087 in the OEIS)), and the numerators produce $1,\,2,\,1,\,3,\,3,\,1,\,2,\,5,\,4,\,4,\ldots$ ((sequence A162910 in the OEIS)).

Properties of the plot of Jimm and the golden ratio

By involutivity, the plot $y=J(x)$ is symmetric about the line $y=x$, and by covariance with $1-x$, it is symmetric about $x+y=1$. The derivative of $J$ vanishes almost everywhere, which can be visually seen from its nearly “flat” segments.

Plot of the Jimm function. Its limit at $0^{+}$ is $1/\varphi$, and at Failed to parse (syntax error): {\displaystyle 1^−} it is $1-1/\varphi$. By involutivity, the value at $1/\varphi$ is 0, and the value at $1-1/\varphi$ is 1. The jump at $x={\tfrac {1}{2}}$ is $1/{\sqrt {5}}$. The function is symmetric about $y=x$ and about $x+y=1$. Its derivative is 0 almost everywhere.

The plot contains many copies of the golden ratio $\varphi$. 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 $\lim _{x\to q^{+}}J(x)$ is some $y={\tfrac {a\varphi +b}{c\varphi +d}}$, where $a,b,c,d\in \mathbf {Z}$ with $ad-bc=\pm 1$. Its Galois conjugate is ${\bar {y}}={\tfrac {a{\bar {\varphi }}+b}{c{\bar {\varphi }}+d}}$, and $J(y)=J({\bar {y}})=q$.

Jimm on real quadratic irrational numbers

Jimm sends real quadratic irrationals to real quadratic irrationals (except the golden ones, which map 2–1 to rationals). It commutes with Galois conjugation: if $J(a+{\sqrt {b}})=A+{\sqrt {B}}$, then $J(a-{\sqrt {b}})=A-{\sqrt {B}}$ (with $a,b,A,B\in \mathbf {Q}$, $b,B>0$ non-squares).

Example: $J{\Bigl (}{\frac {3+5{\sqrt {2}}}{7}}{\Bigr )}\;=\;{\frac {-3+2{\sqrt {95}}}{7}},\quad J({\sqrt {11}})\;=\;{\frac {15+{\sqrt {901}}}{26}}.$

In two-variable form, the functional equations read:

Involutivity: $J(x)=y\iff J(y)=x.$

Covariance in sums: $x+y=1\iff J(x)+J(y)=1.$

Covariance in products: $xy=1\iff J(x)\,J(y)=1.$

Twisted covariance: $x+y=0\iff J(x)\,J(y)=-1.$

From these, we get: ${\tfrac {1}{x}}+{\tfrac {1}{y}}=1\;\;\Longleftrightarrow \;\;{\tfrac {1}{J(x)}}+{\tfrac {1}{J(y)}}=1.$ If $x=a+{\sqrt {b}}\,(b>0)$ is a non-golden real quadratic irrational, then:

1. $N(x)=1\implies N(J(x))=1,$ 2. $\mathrm {Tr} (x)=0\implies N(J(x))=-1,$ 3. $\mathrm {Tr} (x)=1\implies \mathrm {Tr} {\bigl (}J(x){\bigr )}=1,$ 4. $\mathrm {Tr} (1/x)=1\implies \mathrm {Tr} (1/J(x))=1,$

where $N(x)=a^{2}-b$ is the norm and $\mathrm {Tr} (x)=2a$ is the trace. Jimm does not necessarily preserve individual real quadratic fields (it can send two numbers in one field to two numbers in different fields).

Jimm on Markov irrationals

Jimm sends the Markov irrationals to “simpler” quadratic irrationals, e.g.:

Markov number	Markov irrational $x$	$J(x)$
1	${\tfrac {1+{\sqrt {5}}}{2}}$	$\infty$
2	$1+{\sqrt {2}}$	${\sqrt {2}}$
5	${\tfrac {9+{\sqrt {221}}}{10}}$	${\sqrt {6}}-1$
13	${\tfrac {23+{\sqrt {1517}}}{26}}$	${\sqrt {12}}-2$
29	${\tfrac {53+{\sqrt {7565}}}{58}}$	${\sqrt {\tfrac {35}{6}}}-1$
34	${\tfrac {15+5{\sqrt {26}}}{17}}$	${\sqrt {20}}-3$
89	${\tfrac {157+{\sqrt {71285}}}{178}}$	${\sqrt {30}}-4$
194	${\tfrac {86+{\sqrt {21170}}}{97}}$	${\sqrt {\tfrac {119}{10}}}-2$
...	...	...

Jimm and dynamics

Jimm conjugates the Gauss map $G$ (related to the Gauss–Kuzmin–Wirsing operator) to the so-called Fibonacci map $\Phi$: $\Phi \;=\;J\circ G\circ J.$

Since typical real numbers obey the Gauss–Kuzmin distribution, but $J(x)$ accumulates repeated 1’s in its continued-fraction expansion, $J(x)$ breaks that distribution. Thus Jimm sends sets of full measure (those obeying Gauss–Kuzmin) to sets of null measure.

Jimm on higher algebraic numbers

It is widely believed that all algebraic numbers of degree > 2 follow Gauss–Kuzmin statistics (though there is some debate). Under that assumption, $J(x)$ would not follow Gauss–Kuzmin and hence must be transcendental, leading to the conjecture that Jimm sends algebraic numbers of degree > 2 to transcendental numbers. A stronger version asserts any two $J(x)$, $J(y)$ that are algebraically related must lie in the same PGL(2,Z)-orbit (if $x,y$ are algebraic of degree > 2).

Functional equations and equivariant modular forms

For a representation $\rho :PSL_{2}(\mathbf {Z} )\to PGL_{2}(\mathbf {Z} )$, a meromorphic function $h$ on the upper half-plane ${\mathcal {H}}$ is called a $\rho$-covariant function if $h(\gamma \cdot z)=\rho (\gamma )\,h(z)$ for all $\gamma \in PSL_{2}(\mathbf {Z} )$. Equivalently, $h$ is $\rho$-equivariant. It is known that meromorphic covariant functions exist for certain representations, and also that there are meromorphic functions satisfying versions of Jimm’s functional equations.

Some codenumerator values

Below is a table of codenominator values $F{\bigl (}{\tfrac {41}{k}}{\bigr )}$ for various $k$. (Here 41 is chosen arbitrarily.)

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

See also

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