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Draft:Codenominator function

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  • 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)


Codenominator function and the involution Jimm

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The codenominator is a function that extends the Fibonacci sequence to the index set of positive rational numbers, . Many known Fibonacci identities carry over to the codenominator. One can express Dyer's outer automorphism of the extended modular group PGL(2, Z) in terms of the codenominator. The real -covariant modular function Jimm on the real line 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 is related to the arithmetic of real quadratic irrationals.

Definition of the codenominator

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The codenominator function is defined by the following system of functional equations:

 

with the initial condition . The function is called the conumerator. (The name `codenominator' comes from the fact that the usual denominator function can be defined by the functional equations

and the initial condition .)

The codenominator takes every positive integral value infinitely often.

Connection with the Fibonacci sequence

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For integer arguments, the codenominator agrees with the standard Fibonacci sequence, satisfying the recurrence:

The codenominator extends this sequence to positive rational arguments. Moreover, for every rational , the sequence is the so-called Gibonacci sequence [1] (also called the generalized Fibonacci sequence) defined by , and the recursion .

Examples ( is a positive integer)
1
2 , more generally .
3 is the Lucas sequence OEISA000204.
4 is the sequence OEISA001060.
5 is the sequence OEISA022121.
6 is the sequence OEISA022138.
7 is the sequence OEISA061646.
8 , .
9 , .
10 .

Properties [2]

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


2. Fibonacci invariance: For any integer and


3. Symmetry: If , then

 

4. Continued fractions: For a rational number expressed as a simple continued fraction , the value of can be computed recursively using Fibonacci numbers as:

  

5. Reversion:

  

6. Periodicity: For any positive integer , the codenominator is periodic in each partial quotient modulo with period divisible with , where is the Pisano period [3].

7. Fibonacci identities: Many known Fibonacci identities admit a codenominator version. For example, if at least two among are integral, then

where is the codiscriminant[2] (called 'characteristic number' in [1] ). This reduces to Tagiuri's identity[4] when ; which in turn is a generalization of the famous Catalan identity. Any Gibonacci identity[1][5][6] can be interpreted as a codenominator identity. There is also a combinatorial interpretation of the codenominator[7].

The codiscriminant is a 2-periodic function.

Involution Jimm

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

 

This function is involutive and admits a natural extension to non-zero rationals via which is also involutive.

Let be the simple continued fraction expansion of . Denote by the sequence of length . Then:

 

with the rules:

and

.

The function 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 ) is 2-1 valued on golden -or noble- numbers (i.e. the numbers in the PGL(2, Z)-orbit of the golden ratio ).

The extended function

  • sends rationals to rationals[2],
  • sends golden numbers to rationals[2],
  • is involutive except on the set of golden numbers[2],
  • respects ends of continued fractions; i.e. if the continued fractions of has the same end then so does ,
  • sends real quadratic irrationals (except golden numbers) to real quadratic irrationals (see below)[8],
  • commutes with the Galois conjugation on real quadratic irrationals[8] (see below),
  • is continuous at irrationals[8],
  • has jumps at rationals[2],
  • is differentiable a.e.[8],
  • has vanishing derivative a.e., [8]
  • sends a set of full measure to a set of null measure and vice versa[2]

and moreover satisfies the functional equations[8]

Involutivity
   (except on the set of golden irrationals),
Covariance with
   (provided ),
Covariance with
  ,
`Twisted' covariance with
  .

These four functional equations in fact characterize Jimm. Additionally, Jimm satisfies

Reversion invariance
  

Dyer's outer automorphism and Jimm

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

 

where (viewing PGL(2, Z) as a group of Möbius transformations) , and .

The map of generators

  

defines an involutive automorphism PGL(2, Z) PGL(2, Z), called Dyer's outer automorphism[9]. It is known that Out(PGL(2, Z)) is generated by . The modular group PSL(2, Z) PGL(2, Z) is not invariant under . However, the subgroup PSL(2, Z) is -invariant. Conjugacy classes of subgroups of is in 1-1 correspondence with bipartite trivalent graphs, and thus defines a duality of such graphs [10]. This duality transforms zig-zag paths on a graph to straight paths on its -dual graph and vice versa.

Dyer's outer automorphism can be expressed in terms of the codenumerator, as follows: Suppose and . Then

 

The covariance equations above implies that is a representation of as a map P1(R) P1(R), i.e. whenever and PGL(2, Z). Another way of saying this is that is a -covariant map.

In particular, sends PGL(2, Z)-orbits to PGL(2, Z)-orbits, thereby inducing an involution of the moduli space of rank-2 pseudo lattices [11], PGL(2, Z)\ P1(R), where P1(R) is the projective line over the real numbers.

Given P1(R), the involution sends the geodesic on the hyperbolic upper half plane through to the geodesic through , thereby inducing an involution of geodesics on the modular curve PGL(2, Z)\. It preserves the set of closed geodesics because sends real quadratic irrationals to real quadratic irrationals (with the exception of golden numbers, see below) respecting the Galois conjugation on them.

Jimm as a tree automorphism

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Djokovic and Miller[12] constructed as a group of automorphisms of the infinite trivalent tree. In this context, appears as an automorphism of the infinite trivalent tree. is one of the 7 groups acting with finite vertex stabilizers on the infinite trivalent tree [13].

Jimm and the Stern-Brocot tree

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Bird's tree of rational numbers

Applying Jimm to each node of the Stern-Brocot tree permutes all rationals in a row and otherwise preserves each row, yielding a new tree of rationals called Bird's tree, which was first described by Bird[14] . Reading the denominators of rationals on Bird's tree from top to bottom and following each row from left to right gives Hinze's sequence[15]

 (sequence 268087 in the OEIS)

The sequence of conumerators is

 (sequence A162910 in the OEIS)

Properties of the plot of Jimm and the golden ratio

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By involutivity, the plot of is symmetric with respect to the diagonal , and by covariance with , the plot is symmetric with respect to the diagonal . The fact that the derivative of is 0 a.e. can be observed from the plot.

Plot of Jimm. Its limit at 0 + 0+ is 1/φ , and at 1 − 1- it is 1 − 1/φ. By involutivity, the value at 1/φ is 0, and the value at 1 − 1/φ is 1. The amount of jump at x=1/2 is 1/sqrt (5). By involutivity, the plot is symmetric with respect to the diagonal x=y, and by commutativity with 1-x, the plot is symmetric with respect to the diagonal x+y=1. The fact that the derivative of Jimm is 0 a.e. can be observed from the plot.

The plot of Jimm hides many copies of the golden ratio in it. For example

1 ,
2 ,
3 ,
4 ,
5 ,
6

More generally, for any rational , the limit is of the form with and . The limit is its Galois conjugate . Conversely, one has .

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 , then , with and positive non-squares.

For example

 

2-variable form of functional equations: The functional equations can be written in the two-variable form as [16]:

Involutivitiy
Covariance with
Covariance with
Covariance with

As a consequence of these, one has:

If is a real quadratic irrational, which is not a golden number, then as a consequence of the two-variable version of functional equations of one has

1.

2.

3.

4.

where denotes the norm and denotes the trace of .

On the other hand, 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

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Jimm sends the Markov irrationals[17] to 'simpler' quadratic irrationals,[18] see table below.

Markov number Markov irrational
1
2
5
13
29
34
89
169
194
233
433
610
985
1325
1597
2897
4181
5741
6466
7561
9077
10946
14701
28657
33461
37666
43261

Jimm and dynamics

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Jimm conjugates [19] the Gauss map (see Gauss–Kuzmin–Wirsing operator) to the so-called Fibonacci map [20], i.e. .

The expression of Jimm in terms of continued fractions shows that, if a real number obeys the Gauss-Kuzmin distribution, then the asymptotic density of 1's among the partial quotients of is one, i.e. does not obey the Gauss-Kuzmin statistics. For example

21/3=

(21/3)=

This argument also shows that 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

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It is widely believed[21] that if is an algebraic number of degree , then it obeys the Gauss-Kuzmin statistics (for some evidence against this belief, see[22] ). By the above remark, this implies that violates the Gauss-Kuzmin statistics. Hence, according to the same belief, must be transcendental. This is the basis of the conjecture [16] that Jimm sends algebraic numbers of degree to transcendental numbers. A stronger version[23] of the conjecture states that any two algebraically related , are in the same PGL(2, Z)-orbit, if are both algebraic of degree .

Functional equations and equivariant modular forms

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Given a representation , a meromorphic function on is called a -covariant function if

(sometimes is also called a -equivariant function). It is known that[24] there exists meromorphic covariant functions on the upper half plane , i.e. functions satisfying . The existence of meromorphic functions satisfying a version of the functional equations for is also known [2].

Some codenumerator values

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Below is a table of some codenominator values , where 41 is an arbitrarily chosen number.

1 11 21 31
2 12 22 32
3 13 23 33
4 14 24 34
5 15 25 35
6 16 26 36
7 17 27 37
8 18 28 38
9 19 29 39
10 20 30 40

See also

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References

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References

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  1. ^ a b c Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Volume. John Wiley & Sons.
  2. ^ a b c d e f g h Uludağ, A. M.; Eren Gökmen, B. (2022). "The conumerator and the codenominator". Bulletin des Sciences Mathématiques. 180 (180): 1–31. doi:10.1016/j.bulsci.2022.103192. PMID 103192.
  3. ^ 'Pisano' is another name of Fibonacci
  4. ^ A. Tagiuri, Di alcune successioni ricorrenti a termini interi e positivi, Periodico di Matematica 16 (1900–1901), 1–12.
  5. ^ Some Weighted Generalized Fibonacci Number Summation Identities, Part 1, arXiv:1903.01407
  6. ^ Some Weighted Generalized Fibonacci Number Summation Identities, Part 2, arXiv:1903.01407
  7. ^ Mahanta, P. J., & Saikia, M. P. (2022). Some new and old Gibonacci identities. Rocky Mountain Journal of Mathematics, 52(2), 645-665.
  8. ^ a b c d e f Uludağ, A. M.; Ayral, H. (2019). "An involution of reals, discontinuous on rationals, and whose derivative vanishes ae". Turkish Journal of Mathematics. 43 (3): 1770–1775. doi:10.3906/mat-1903-34.
  9. ^ Dyer, J. L. (1978). "Automorphic sequences of integer unimodular groups". Illinois Journal of Mathematics 22 (1) 1-30.
  10. ^ Jones, G. A., & Singerman, D. (1994). Maps, hypermaps and triangle groups. The Grothendieck Theory of Dessins d’Enfants (L. Schneps ed.), London Math. Soc. Lecture Note Ser, 200, 115-145.
  11. ^ Manin YI (2004). Real multiplication and noncommutative geometry (ein Alterstraum). In the Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, (pp. 685-727). Berlin, Heidelberg: Springer Berlin Heidelberg.
  12. ^ D. Z. Djokovic, D.G. L. MILLER (1980), Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B 29 (1980) 195-230.
  13. ^ Conder, M., & Lorimer, P. (1989). Automorphism groups of symmetric graphs of valency 3. Journal of Combinatorial Theory, Series B, 47(1), 60-72.
  14. ^ R.S. Bird (2006) Loopless functional algorithms, in: International Conference on Mathematics of Program Construction, Jul 3, Springer, Berlin, Heidelberg, pp. 90–114.
  15. ^ R. Hinze (2009), The Bird tree, J. Funct. Program. 19 (5) 491–508.
  16. ^ a b Uludag, A.M. and Ayral, H. (2021) On the involution Jimm. Topology and geometry–a collection of essays dedicated to Vladimir G. Turaev, pp.561-578.
  17. ^ Aigner, Martin (2013). Markov's theorem and 100 years of the uniqueness conjecture : a mathematical journey from irrational numbers to perfect matchings. New York: Springer. ISBN 978-3-319-00887-5. OCLC 853659945.
  18. ^ B. Eren, Markov Theory and Outer Automorphism of PGL(2,Z), Galatasaray University Master Thesis, 2018.
  19. ^ Uludağ, A. M.; Ayral, H. (2018). "Dynamics of a family of continued fraction maps". Dynamical Systems. 33 (3): 497–518. doi:10.1080/14689367.2017.1390070.
  20. ^ C. Bonanno and S. Isola. (2014). " A thermodynamic approach to two-variable Ruelle and Selberg zeta functions via the Farey map", Nonlinearity. 27 (5) 10.1088/0951-7715/27/5/897
  21. ^ Bombieri, E. and van der Poorten, A. (1975): “Continued Fractions of Algebraic Numbers”, in: Baker (ed.), Transcendental Number Theory, Cambridge University Press, Cambridge, 137-155.
  22. ^ Sibbertsen, Philipp; Lampert, Timm; Müller, Karsten; Taktikos, Michael (2022), Do algebraic numbers follow Khinchin's Law?, arXiv:2208.14359
  23. ^ Testing the transcendence conjectures of a modular involution of the real line and its continued fraction statistics, Authors: Hakan Ayral, A. Muhammed Uludağ, arXiv:1808.09719
  24. ^ Saber, H., & Sebbar, A. (2022). Equivariant solutions to modular Schwarzian equations. Journal of Mathematical Analysis and Applications, 508(2), 125887