aa r X i v : . [ m a t h . QA ] O c t ON q -DEFORMED REAL NUMBERS SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
Abstract.
We associate a formal power series with integer coefficients to a positive real number, weinterpret this series as a “ q -analogue of a real.” The construction is based on the notion of q -deformedrational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers,we obtain certain Laurent series. Introduction
We take a new and experimental route to introduce a certain version of “ q -deformed real numbers”,extending q -deformations of rationals introduced in [6]. Given a real number x ≥
0, we will construct aformal power series with integer coefficients associated with x . There is no explicit formula to determinethe coefficients of this series, but there is an algorithm to calculate them.Our construction is completely different from the classically known q -deformations of a number x ∈ R ,defined by the formulas q x − q − (or q x − q − x q − q − ) that do not give power series with integer coefficients. The onlycase where our construction coincides with the classical one is that of integers. Throughout this paper,we always use the Gauss definition: [ a ] q = q a − q − , a ∈ Z , that gives the familiar polynomials [ a ] q = 1 + q + · · · + q a − and [ − a ] q = − q − − q − − · · ·− q − a , for a ∈ N .All our constructions will be in accordance with these formulas.It is of course too early to discuss possible applications of the q -deformed real numbers, since we donot know sufficiently general properties of the power series we obtain. However, the existence of theprocedure is quite surprising and the examples are captivating.The q -deformation of a rational number rs is a quotient of two polynomials: h rs i q = R ( q ) S ( q ) , where R and S both depend on r and s (see [6]). In this paper, we represent these rational functions asTaylor series at q = 0.The definition of q -reals is as follows. Let first x ≥ x n ) n ≥ anysequence of rational numbers that converges to x . We q -deform the sequence ( x n ) n ≥ to obtain a sequenceof rational functions: [ x ] q , [ x ] q , [ x ] q , . . . For every n ≥
1, we consider the Taylor expansion of therational function [ x n ] q at q = 0:(1) [ x n ] q =: X k ≥ κ n,k q k . Key words and phrases. q -rationals, q -continued fractions, total positivity, Farey tessellation. SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
Abusing the notation, we use the same name, [ x n ] q , for the Taylor series. The q -deformation of x is theseries(2) [ x ] q := X k ≥ κ k q k , where κ k = lim n →∞ κ n,k . The existence of the limit and its independence of the choice of the converging sequence ( x n ) n ≥ isguaranteed by the following theorem. Theorem 1.
Given an irrational real number x ≥ , for every k ≥ the coefficients κ n,k of the Taylorseries (1) stabilize as n grows. Moreover, the limit coefficients κ k in (2) are integers that do not dependon the choice of the converging sequence of rationals ( x n ) n ≥ . This statement was first observed by computer experimentation, but then a simple proof was found.Note that the coefficients of the polynomials in the numerator and denominator of the sequence of rationalfunctions [ x n ] q do not stabilize. They grow with n infinitely at every fixed power of q .In practice, we always construct q -deformed real numbers using continued fractions. Let x ≥ x = [ a , a , a , . . . ] , where a i ’s are positive integers, its continued fraction expansion.The sequence of rational numbers x n := [ a , . . . , a n ]approximates x ; it is called the sequence of convergents. In this case the stabilization phenomenon ofTheorem 1 can be controlled with a greater exactness. Proposition 1.1.
Let x ≥ be an irrational real number. The Taylor expansions at q = 0 of twoconsecutive q -deformed convergents of the continued fraction of x , namely of x n − = [ a , . . . , a n − ] and x n = [ a , . . . , a n ] , have the first a + · · · + a n − terms identical, the coefficients of q a + ··· + a n − differby . With the help of a computer program, we carried out a number of tests calculating q -deformationsof known mathematical constants, from the simplest golden ratio to the transcendental e and π , andchecking their properties. The most pleasant surprise for us was the appearance of the sequence ofgeneralized Catalan numbers (sequence A004148 of [7]) as coefficients κ k of the deformed golden ratio.This remarkable “coincidence” and known properties of A004148 allowed us to conjecture (and eventuallyprove) several properties of quadratic irrationals.We know only very few general properties of q -deformed real numbers. One of them is the action ofthe translation group Z described by following formula(3) [ x + 1] q = q [ x ] q + 1 , [ x − q := [ x ] q − q . Note that the second equation in (3) allows us to extend our q -deformations to the case x < Theorem 2. If k ≤ x ≤ k + 1 , where k ∈ Z > , then the k -th order coefficient of the series [ x ] q vanishes,while all the preceding coefficients are equal to : [ x ] q = 1 + q + q + · · · + q k − + κ k +1 q k +1 + κ k +2 q k +2 + · · · Theorem 2 implies that for negative x we obtain Laurent series instead of power series. More precisely,if − k ≤ x < − k , where k ∈ Z > , then [ x ] q is of the following general form(4) [ x ] q = − q − k + κ − k q − k + κ − k q − k + · · · where κ i ∈ Z . -DEFORMED REAL NUMBERS 3 Let us sum up our understanding of q -deformation, or “quantization”, of real numbers in comparisonwith integers and rationals. This quantization transforms integers into polynomials, rationals into rationalfractions, and real numbers into power series, in each case with integer coefficients:[ . ] q : Z ≥ −→ Z ≥ [ q ] , Q ≥ −→ Z ≥ ( q ) , R ≥ −→ Z [[ q ]] , R −→ Z [[ q ]][ q − ] . In the case of integers and rationals, the resulting polynomials and rational functions are with positivecoefficients. In the case of real numbers, this positivity consists in the fact that the power series areobtained as limit of rationals functions with positive integer coefficients.The paper is organized as follows.In Section 2 we briefly recall the notion of q -rational. We start with the most elementary, recurrentway to calculate q -rationals, and then give two equivalent and more explicit formulas. The first oneuses the continued fractions and the second 2 × q -deformed quadraticirrational numbers. We start with the golden ratio and continue with the “silver ratio” and severalexamples of square roots, giving in each case an explicit formula of the q -deformation.In Section 5 we consider two examples of transcendental irrationals, namely e and π . We calculate thefirst terms of their q -deformations trying to make some observations.In Section 6 we discuss the action of the translation group and prove Theorem 2.It would be interesting to investigate more concrete examples. Note that we searched in vain for somefunctional equations similar to those of q -deformed quadratic irrational numbers in the case of higherorder algebraic numbers, such as √
2. 2. q -deformed rationals In this section, we try to give a transparent and self-contained exposition of the notion of q -rationalintroduced in [6]. We outline an analogy with q -binomial coefficients, and give a recurrent way to compute q -rationals from the Farey graph. Finally we give two explicit formulas for the q -rationals using continuedfraction expansions and the matrix form.2.1. Analogy with the q -binomials. Recall that the classical Gaussian q -binomial coefficients (see [8])are polynomials in q that can be calculated recurrently via the formula(5) (cid:18) rs (cid:19) q = (cid:18) r − s − (cid:19) q + q s (cid:18) r − s (cid:19) q . SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
The q -binomial coefficients are the vertices of the “weighted Pascal triangle” which encodes the aboveformula: (cid:0) (cid:1) q ✹✹✹✹ ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q ✹✹✹✹ q ✡✡✡✡ (cid:0) (cid:1) q (cid:0) (cid:1) q (cid:0) (cid:1) q (cid:0) (cid:1) q (cid:0) (cid:1) q · · · · · · The idea behind the definition of q -rationals is to use exactly the same rule, but replace the Pascal triangleby the Farey graph.2.2. The weighted Farey graph.
The structure of the Farey graph is as follows (see [4]). The set ofvertices consists of rational numbers Q , completed by ∞ := . Two rationals, rs and r ′ s ′ (always writtenas irreducible fractions), are connected by an edge if and only if rs ′ − r ′ s = ±
1. Edges of the Farey graphare often represented as (non-crossing) geodesics of the hyperbolic plane.Although the Farey graph is much more complicated than the Pascal triangle, in particular everyvertex rs has infinitely many neighbors, these two graphs have one property in common: each vertex hastwo “parents”. In the Farey graph these “parents” are characterized as follows. Among the infinite setof neighbors of rs there are exactly two, r ′ s ′ and r ′′ s ′′ , that are also connected to each other. In other words,every rational rs belongs to exactly one triangle r ′ s ′ rs r ′′ s ′′ such that r ′ s ′ < rs < r ′′ s ′′ . Furthermore, one has rs = r ′ + r ′′ s ′ + s ′′ .Similarly to the case of q -binomials, the edges of the weighted Farey graph are labeled by powers of q according to the following pattern 1 q ℓ q ℓ − h r ′ s ′ i q (cid:2) rs (cid:3) q h r ′′ s ′′ i q The vertices are labeled by the following rule: if h r ′ s ′ i q = R ′ S ′ and h r ′′ s ′′ i q = R ′′ S ′′ , then(6) h rs i q := R ′ + q ℓ R ′′ S ′ + q ℓ S ′′ . Note that (6) is analogous to (5).The weights q ℓ and the q -rationals can be calculated recursively along the Farey graph; see Figure 1. -DEFORMED REAL NUMBERS 5 b b b bbbbb b b b b b b b b b b b b b q q q q q q q q −
101 11 (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q q q q q q q q q (cid:2) (cid:3) q (cid:2) (cid:3) q (cid:2) (cid:3) q Figure 1.
Upper part of the weighted Farey graph between and For instance, (cid:2) (cid:3) q = q +2 q +2 q + q q +2 q + q . We will be interested by the corresponding Taylor series at q = 0.For instance, for the above example the series starts as follows (cid:20) (cid:21) q = 1 + q − q + q + 3 q − q − q + 7 q + 4 q − q ± · · · q -deformed continued fractions. We will now give another method of computing q -rationals. Itis relied on the continued fraction expansion.Given a rational number rs > r and s are positive integers assumed to be coprime. It has aunique continued fraction expansion with even number of terms(7) rs = a + 1 a + 1 . . . + 1 a m , with a i ≥
1, denoted by [ a , . . . , a m ]. Note that the choice of even length removes the ambiguity[ a , . . . , a n ,
1] = [ a , . . . , a n + 1] and makes the expansion unique. SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
In order to calculate the q -deformation (cid:2) rs (cid:3) q , one can use the following explicit formula. Given aregular continued fraction [ a , . . . , a m ], its q -deformation is given by(8) [ a , . . . , a m ] q := [ a ] q + q a [ a ] q − + q − a [ a ] q + q a [ a ] q − + q − a . . . [ a m − ] q + q a m − [ a m ] q − where we use the standard notation for the q -integers [ a ] q = 1 + q + q + · · · + q a − . Of course, one canget rid of negative exponents in (8), but the formula becomes uglier.2.4. The matrix formulas.
We give another, equivalent, form to define q -rationals. It uses 2 × rs be a rational written in the form of a continued fraction expansion (7). Consider the2 × f M + q ( a , . . . , a m ) in [6]. It is defined bythe formula(9) f M + q ( a , . . . , a m ) := [ a ] q q a ! q [ a ] q q a ! · · · [ a m − ] q q a m − ! q [ a m ] q q a m ! . This matrix is a q -analogue of the usual “matrix of convergents” of a continued fraction (see [2, 5]).It is easy to prove that this matrix contains the numerator and denominator of (cid:2) rs (cid:3) q (up to multipli-cation by q ) in the first column:(10) f M + q ( a , . . . , a m ) = q R R ′ m − q S S ′ m − ! , where R ( q ) S ( q ) = (cid:2) rs (cid:3) q , and where R ′ m − ( q ) S ′ m − ( q ) = [ a , . . . , a m − ] q is the previous convergent.3. The stabilization phenomenon
In this section we first collect some basic properties that follow from the matrix presentation. Weprove Proposition 1.1 and Theorem 1. Finally, in a remark, we discuss the stabilization phenomenon inthe case when x is rational.3.1. Some simple properties of q -rationals. The following statements are immediate corollaries of (9)and (10).Let rs ≥ (cid:2) rs (cid:3) q = R ( q ) S ( q ) , where R ( q ) and S ( q ) are polynomials with positivecoefficients whose highest and lowest coefficients are equal to 1. The degrees of the polynomials R and S are as follows deg( R ) = a + . . . + a m − , deg( S ) = a + . . . + a m − . The unimodality conjecture of [6] states that the coefficients of R and S first grow and then decreasemonotonically.Let us mention that the most important properties of q -rationals are the total positivity and thecombinatorial interpretation of the coefficients of R and S . -DEFORMED REAL NUMBERS 7 Proof of Proposition 1.1.
Let x ≥ x n − and x n be two consecutiveconvergents of its continued fraction. Let[ x n − ] q = R n − S n − and [ x n ] q = R n S n , One then has tautologically R n S n − R n − S n − = R n S n − − S n R n − S n S n − . The polynomial in the numerator of the right-hand-side is a power of q :(11) R n S n − − S n R n − = q a + ··· + a n − . Indeed, to prove this, it suffices to compare the determinant of the matrix f M + q ( a , . . . , a m ) written inthe forms (9) and (10).Both polynomials, S n and S n − , start with zero-order term 1, and so is the series 1 / ( S n S n − ). Itfollows now from (11), that the series [ x n ] q − [ x n − ] q is of the form[ x n ] q − [ x n − ] q = q a + ··· + a n − + O ( q a + ··· + a n ) . Hence, Proposition 1.1.3.3.
Proof of Theorem 1.
Let now ( y n ) n ≥ be an arbitrary sequence of rationals converging to x .Then, for every fixed m , there exists N such that y n ∈ [ x m − , x m ] (or y n ∈ [ x m , x m − ]) for every n ≥ N .Here, as above, ( x n ) n ≥ is the sequence of convergents of the continued fraction of x .By Proposition 1.1, the first a = a + · · · + a m − x m − ] q and [ x m ] q coincide. It turns out that the same is true for every rational between x m − and x m . Lemma 3.1.
For every rational rs such that, x m − < rs < x m , the first a = a + · · · + a m − terms ofthe Taylor series of (cid:2) rs (cid:3) q coincides with those of [ x m − ] q and [ x m ] q .Proof. Let [ x m − ] q = R m − S m − and [ x m ] q = R m S m . Recall that x m − and x m are joined by an edge in theFarey graph.Suppose first that rs is also joint to x m − and x m , so that we have a triangle1 q ℓ q ℓ − R m − S m − (cid:2) rs (cid:3) q R m S m Set (cid:2) rs (cid:3) q = RS , then, by definition of q -rationals, RS = R m − + q ℓ R m S m − + q ℓ S m By (11), we have R m S m − − S m R m − = q a . Therefore,(12) RS m − − SR m − = R m − S m − + q ℓ R m S m − − R m − S m − − q ℓ R m − S m = q a + ℓ RS m − SR m = R m − S m + q ℓ R m S m − R m S m − − q ℓ R m S m = − q a . Using the same argument as in the proof of Proposition 1.1 we deduce the statement of the lemma inthe case where the three points form a triangle.
SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
The general case of the lemma can then be proved inductively. Indeed, every rational rs such that, x m − < rs < x m can be joined to x m − and x m by a sequence of triangles. To see this, draw a verticalline in the Poincar´e half-plane through rs and collect all the triangles of the Farey tessellation between x m − and x m crossed in their interior by this line.Hence, the lemma. (cid:3) Theorem 1 follows from Lemma 3.1 and Proposition 1.1.
Remark 3.2.
We also investigated the stabilization phenomenon in the case when x is rational. If x = rs and (cid:16) r n s n (cid:17) n ≥ is a sequence converging to x it is natural to ask whether [ x ] q defined by (2) is equal tothe q -rational (cid:2) rs (cid:3) q . The answer is surprising: when the sequence (cid:16) r n s n (cid:17) approaches rs from the right thestabilized power series defined by (2) is equal to the q -rational (cid:2) rs (cid:3) q , when the sequence approaches therational from the left this is no longer true. This is due to the asymmetry of the relations (12).Indeed, if r n s n > rs for all n >>
0, one can use the same arguments as in the proof of Theorem 1 byreplacing the sequence of convergents x n by the sequence of right neighbors nr + r ′′ ns + s ′′ , where r ′′ s ′′ is the rightparent introduced in Section 2.2. Considering Taylor expansions, the right neighbors h nr + r ′′ ns + s ′′ i q have moreand more first terms identical to those of (cid:2) rs (cid:3) q as n grows.If r n s n < rs for all n >> h r n s n i q , but the stabilized powerseries is different from (cid:2) rs (cid:3) q . For instance, testing several sequences of rational approaching the integer 2with smaller values we always obtained a stabilization to the series 1 + q which is not [2] q = 1 + q .4. q -deformations of quadratic irrationals In this section we discuss several examples of quadratic irrational numbers. We start from the simplestpossible case of the golden ratio and identify the coefficients of the Taylor series as the remarkable andthoroughly studied sequence of generalized Catalan numbers. We dwell on this first example with moredetails to better explain the stabilization phenomenon. We then calculate the q -deformation of thenumber 1 + √
2, which is usually called the “silver ratio”. Finally, we consider several examples of squareroots of small positive integers and calculate the corresponding functional equations.4.1.
The golden ratio and generalized Catalan numbers.
The simplest example of an infinitecontinued fraction is the expansion of the golden ratio: ϕ = 1 + √
52 = [1 , , , , , . . . ] . The convergents are ratios of consecutive Fibonacci numbers: ϕ n = F n +1 /F n .According to (8), the q -deformation of ϕ is given by the 2-periodic infinite continued fraction(13) [ ϕ ] q = 1 + q q + 11 + q q + 1 . . . Remark 4.1.
Let us mention that there exists a celebrated q -deformation of the golden ratio, calledthe Rogers-Ramanujan continued fraction. It is aperiodic and has a great number of beautiful and -DEFORMED REAL NUMBERS 9 sophisticated properties (for a survey, see [1]). Unlike the Rogers-Ramanujan continued fraction, we donot know if (13) is a quotient of two q -series with positive coefficients.Consider the q -deformations of the convergents [ ϕ n ] q . We proved in [6] that the coefficients of thepolynomials of [ ϕ n ] q are the numbers appearing in the remarkable “Fibonacci lattice” (see A123245 ofOEIS [7] and its mirror A079487). More precisely, A123245 appears in the numerator and A079487 inthe denominator. For instance,[ ϕ ] q = 1 + 2 q + 3 q + 3 q + 3 q + q q + 2 q + 2 q + q , [ ϕ ] q = 1 + 3 q + 5 q + 7 q + 7 q + 6 q + 4 q + q q + 4 q + 5 q + 4 q + 3 q + q , [ ϕ ] q = 1 + 4 q + 7 q + 10 q + 11 q + 10 q + 7 q + 4 q + q q + 6 q + 7 q + 7 q + 5 q + 3 q + q . We see that the coefficients at every fixed power of q of the numerator and the denominator grow. Toillustrate the stabilization phenomenon, we give the corresponding Taylor series:[ ϕ ] q = 1 + q − q + 2 q − q + 3 q − q + 4 q − q + 5 q − q + 6 q · · · [ ϕ ] q = 1 + q − q + 2 q − q + 8 q − q + 30 q − q + 103 q − q + 368 q · · · [ ϕ ] q = 1 + q − q + 2 q − q + 8 q − q + 37 q − q + 184 q − q + 932 q · · · The series [ ϕ ] approximates [ ϕ ] q correctly up to the 9th term, while [ ϕ ] only up to q .The full series (13) starts as follows:[ ϕ ] q = 1 + q − q + 2 q − q + 8 q − q + 37 q − q + 185 q − q + 978 q − q +5373 q − q + 30372 q − q + 175502 q − q + 1032004 q · · · Note that one needs the n th convergent [ ϕ n ] q to calculate [ ϕ ] q with accuracy up to q n .Fix the notation [ ϕ ] q =: X k ≥ φ k q k . We were able to identify the coefficients φ k appearing in this series as the so-called Generalized Catalannumbers (see sequence A004148 of OEIS), but with alternating signs. Proposition 4.2.
One has φ k = ( − k a k − , for k ≥ , where a k are the Generalized Catalan numbers;see A004148 of [7] .Proof. Let us prove that the series [ ϕ ] q satisfies the following functional equation:(14) q [ ϕ ] q − (cid:0) q + q − (cid:1) [ ϕ ] q − , which is a q -analogue of ϕ = ϕ + 1. In fact, (14) is immediate consequence of (13) which can also bewritten [ ϕ ] q = 1 + q q + 1[ ϕ ] q . Proposition 4.2 then follows from the known results about the generating function of the GeneralizedCatalan numbers. Indeed, this generating function satisfies an equation equivalent to (14). (see M.Somos’ contribution to A004148). (cid:3) This observation is due to Doron Zeilberger.
Solving (14), one obtains [ ϕ ] q = q + q − p q + 2 q − q + 2 q + 12 q . Obviously, at q = 1 one recovers the golden ratio ϕ . Remark 4.3.
Let us also mention that the coefficients of [ ϕ ] q satisfy, for all n ≥
3, the following linearrecurrence ( k + 1) φ k + (2 k − φ k − + (2 − k ) φ k − + (2 k − φ k − + ( k − φ k − = 0 . In the context of the generalized Catalan numbers, this recurrence was conjectured by R.J. Mathar in2011, and recently proved; see [3] (based on [9]).4.2.
The q -deformed silver ratio. The number 1 + √ , , , , . . . ] is often called the silver ratio.It is denoted by δ S , and its convergents are given by the quotient of consecutive Pell numbers. This isprobably the next simplest example of infinite continued fraction after the golden ratio.Formula (8) implies the following q -deformation(15) [ δ S ] q = 1 + q + q q + q + 11 + q + q q + q + 1 . . . The stabilization process goes twice faster than for the golden ratio: one needs the n th convergent tocalculate [ δ S ] q with accuracy up to q n . The series [ δ S ] q starts as follows[ δ S ] q = 1 + q + q − q + q + 4 q − q − q + 18 q + 7 q − q + 18 q +146 q − q − q + 692 q + 476 q − q + 307 q +7322 q − q − q + 33061 q + 33376 q − q − q · · · We see that the coefficients grow much slower than those of [ ϕ n ] q . This sequence of coefficients is notin OEIS. Proposition 4.4.
The series [ δ S ] q satisfies the following functional equation: (16) q [ δ S ] q − (cid:0) q + 2 q − (cid:1) [ δ S ] q − . Proof.
Fromula (15) rewritten in the form[ δ S ] q = 1 + q + q q + q + 1[ δ S ] q readily implies (16). (cid:3) Equation (16) is a q -analogue of δ S = 2 δ S + 1, appearance of q in this formula is somewhat surprising.Solving (16), one obtains [ δ S ] q = q + 2 q − p q + 4 q − q + 4 q + 12 q . -DEFORMED REAL NUMBERS 11 The q -square roots of , , and . We calculate the q -analogues of the simplest square roots.Recall that √ , , √ , , , √ , , √ , , , , (cid:2) √ (cid:3) q = 1 + q − q + q + 4 q − q − q + 18 q + 7 q − q + 18 q +146 q − q − q + 692 q + 476 q − q + 307 q +7322 q − q − q + 33061 q + 33376 q · · · (cid:2) √ (cid:3) q = 1 + q − q + 2 q − q − q + 7 q − q + 7 q + 18 q − q + 78 q − q − q + 514 q − q − q + 2591 q − q +1837 q + 9405 q − q + 33390 q + 10329 q · · · (cid:2) √ (cid:3) q = 1 + q + q − q − q − q + 3 q + 4 q − q − q − q + 2 q +25 q + 22 q − q − q − q + 67 q + 208 q + 168 q − q · · · (cid:2) √ (cid:3) q = 1 + q + q − q + 2 q − q + 4 q − q + 8 q − q + 9 q − q − q +40 q − q + 215 q − q + 724 q − q + 1845 q − q + 3324 q − q + 1696 q + 4157 q · · · Note that the coefficients of (cid:2) √ (cid:3) q are those of the silver ratio, but the power of q is shifted by 1; in fullaccordance with (3).The following formulas can be proved in a similar way as (14) and (16), The calculations are quitelong but straightforward, so we omit the details. Proposition 4.5.
The series (cid:2) √ (cid:3) q , (cid:2) √ (cid:3) q , (cid:2) √ (cid:3) q and (cid:2) √ (cid:3) q satisfy the following functional equations: q h √ i q − (cid:0) q − (cid:1) h √ i q = q + 1 , (17) q h √ i q − (cid:0) q + q − q − (cid:1) h √ i q = q + q + 1 , (18) q h √ i q − ( q + q − q − h √ i q = q + q + q + q + 1 , (19) q h √ i q − ( q + q − q − h √ i q = q + 2 q + q + 2 q + 1 . (20)We obtain, consequently, the following expressions for the quantized square roots. (cid:2) √ (cid:3) q = q − p q + 4 q − q + 4 q + 12 q , (cid:2) √ (cid:3) q = q + q − q − p q + 2 q + 3 q + 3 q + 2 q + 12 q , (cid:2) √ (cid:3) q = q + q − q − p q + 2 q + 2 q + 5 q + 5 q + 2 q + 2 q + 12 q , (cid:2) √ (cid:3) q = q + q − q − p q + 2 q + q + 4 q + 6 q + 6 q + 4 q + q + 2 q + 12 q . Note that (cid:2) √ (cid:3) q looks quite different from the golden ratio. This is an example of a highly non-trivialaction of homothety x → x/ q -deformations ofarbitrary quadratic irrationals. q -deformations of e and π In this section we write down the first terms of the q -deformations of two notable examples of tran-scendental irrational numbers, e and π . We calculated several hundreds of terms to convince ourselvesthat the coefficients of the corresponding series do not correspond to any sequence of OEIS. However,one can make some surprising observations.5.1. Computing [ e ] q . The continued fraction expansion of the Euler constant is given by the followingfamous regular pattern e = [2 , , , , , , , , , , , , , , , . . . ] (see sequence A003417 in the OEIS).To calculate the first 40 terms in the series [ e ] q , one needs to take the 15th convergent e = [2 , , , , , , , , , , , , , ,
10] = 517656 / . The series [ e ] q starts as follows:[ e ] q = 1 + q + q − q + 2 q − q + 3 q − q − q + 9 q − q + 25 q − q + 23 q + 2 q − q + 134 q − q + 320 q − q + 243 q + 71 q − q + 1531 q − q + 3504 q − q + 2747 q + 488 q − q + 15395 q − q + 34716 q − q + 24771 q + 9096 q − q + 156811 q · · · We observe that the coefficients of q k , where k ≥
0, turn out to be smaller than those of their neighbors.The signs of the coefficients also obey a certain 7-periodic pattern: indeed, the double plus, i.e., “+ , +”signs, appears with period 7. We do not know any reason for such a “7-periodicity” related to Euler’snumber.5.2. The quantum π . The continued fraction expansion of π (cf. sequence A001203 in the OEIS)starts as follows: π = [3 , , , , , , , , , , , , , , , , , , , , , , . . . ]. The rules governingthis sequence are unknown.The 4th convergent π = [3 , , ,
1] = 355 /
113 gives 24 first terms of [ π ] q , and already the 5thconvergent π = [3 , , , , / π ] q up to degree 317. Wecalculated up to [ π ] q = 11895062545096656711950 / π upto 43 digits, and that gives the first 603 terms of the series [ π ] q . The first 79 terms of [ π ] q are:[ π ] q = 1 + q + q + q − q − q + q + q − q − q − q + 2 q + 4 q + q − q − q − q + q + 5 q + 8 q + 3 q − q − q − q − q + 8 q + 19 q + 20 q + 2 q − q − q − q + 31 q + 51 q + 45 q − q − q − q − q + 35 q + 122 q + 140 q + 72 q − q − q − q − q + 171 q + 383 q + 363 q + 76 q − q − q − q − q + 702 q + 1101 q + 790 q − q − q − q − q + 642 q + 2454 q + 2982 q + 1415 q · · · The coefficients of this series grow very slowly in contrast with the other examples we considered sofar. Asymptotic of the ratio of two consecutive coefficients seems to be close to 1, that would implythat the radius of convergence of the series is equal to 1. A curious observation is that, for unknownreasons, the coefficient of q vanishes. One also observes oscillation of the sequence of coefficients, and -DEFORMED REAL NUMBERS 13 the unimodality property of every (short) subsequence of coefficients with constant sign. We were unableto find any pattern or to conjecture any functional equation for this series.6. Translations
In this section we consider the properties of q -deformed reals under translations of the argument. Theaction of the translation group Z is defined by the operator T : x → x + 1 and its inverse, T − : x → x − T which is simpler, and then T − brings us to a new ground. In particular, we extend thenotion of q -deformation to negative real numbers.6.1. Right translations.
Consider first the action of T . Proposition 6.1. If x ≤ , then [ x + 1] q = q [ x ] q + 1 .Proof. The statement of the proposition is obvious when x is a q -integer. Hence by the explicit formula (8)the statement is also clear for a rational x . The irrational case will then follow from the stabilizationphenomenon. (cid:3) The coefficient “gap”.Proposition 6.2.
The first order term of series [ x ] q vanishes: [ x ] q = 1 + κ q + κ q + · · · if and only if ≤ x ≤ .Proof. As in the previous proof, it suffices to prove the statement for a rational x . Let rs = [ a , a , . . . , a m ]be a rational written in a form of a continued fraction. It is an easy computation that the polynomials R = 1 + r q + · · · and S = 1 + s q + · · · of the q -deformed rational (cid:2) rs (cid:3) q = RS have identical first-ordercoefficient: r = s if and only if a = 1.The result then follows from the formula RS = 1 + r q + · · · s q + · · · = (1 + r q + · · · )(1 − s q + · · · ) . (cid:3) Propositions 6.1 and 6.2 together imply Theorem 2.6.3.
Left translations and q -deformed negative numbers. The heuristic definition[ x − q := [ x ] q − q , that we adopt now, leads to a classes of q -deformed reals that we did not consider so far. For x ≥
2, theaction of T − is a tautological inverse of Proposition 6.1. The situation changes for 1 ≤ x ≤ ≤ x <
1, then the zero-order term of [ x ] q vanishes:[ x ] q = κ q + κ q + κ q + · · · (2) If − ≤ x <
0, then [ x ] q = − q + κ + κ q + κ q + κ q + · · · (3) More generally, applying the operator T − several times, we obtain the general form (4) of [ x ] q in the case − k ≤ x < − k . Example 6.3. (a) If n is a positive integer, then [ − n ] q = q − n + q − n + · · · + q − .(b) For x = we have (cid:2) (cid:3) q = q q , and for x = − we have (cid:2) − (cid:3) q = − q (1+ q ) , so that (cid:20) (cid:21) q + (cid:20) − (cid:21) q = − q + 1 . We calculated the series corresponding to the negations of several examples of quadratic irrationals,and observed the following property. Some of them satisfy the property that the sum of the series[ x ] q + [ − x ] q contains only finitely many terms. Example 6.4. (a) For −√
2, we have h −√ i q = − q − q − q + 2 q − q − q + 5 q + 7 q − q − q + 55 q − q · · · Starting from the third-order term, this series is the negation of (cid:2) √ (cid:3) q .(b) For −√
7, we have h −√ i q = − q − q − q − q + q − q + 3 q − q + 6 q − q + 9 q − q + 5 q + 9 q · · · Once again, starting from the third-order term, the series is the negation of (cid:2) √ (cid:3) q .This property is a mystery to us, since, as we can see already for x = , it fails to be true even forrationals.Note also, that nothing similar to Example 6.4 happens for more sophisticated irrationals. Example 6.5.
For − π , we have the series that starts as follows[ − π ] q = − q − q − q − q + q − q + q − q + 2 q − q + 3 q − q − q + 3 q − q +6 q − q − q + 4 q − q + 10 q − q + 3 q − q − q + 5 q − q · · · and seems to have nothing in common with [ π ] q . Acknowledgements . We are grateful to Vladlen Timorin for a suggestion to look at Taylor coefficients.We are also grateful to Fr´ed´eric Chapoton for a SAGE program computing q -rationals and to DoronZeilberger for the proof of the functional equations formulated as conjectures in the first version of thispaper. This paper was partially supported by the ANR project SC3A, ANR-15-CE40-0004-01. References [1] B. Berndt, H. Chan, S-S. Huang, S-Y. Kang, J. Sohn, S. Son,
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Automated Proofs of Many Conjectured Recurrences in the OEIS made by R.J.Mathar , arXiv:1707.04654.[4] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers. Sixth edition. Revised by D. R. Heath-Brownand J. H. Silverman. With a foreword by Andrew Wiles. Oxford University Press, Oxford, 2008, 621 pp.[5] S. Morier-Genoud, V. Ovsienko,
Farey boat. Continued fractions and triangulations, modular group and polygon dis-sections , Jahresber. Dtsch. Math.-Ver. (2019), no. 2, 91–136.[6] S. Morier-Genoud, V. Ovsienko, q -deformed rationals and q -continued fractions , arXiv:1812.00170.[7] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org.[8] R. Stanley, Enumerative combinatorics. Volume 1. Second edition. Cambridge Studies in Advanced Mathematics, 49.Cambridge University Press, Cambridge, 2012. xiv+626 pp.[9] D. Zeilberger, The C-finite ansatz , Ramanujan J. (2013), no. 1-2, 23–3. -DEFORMED REAL NUMBERS 15 Sophie Morier-Genoud, Sorbonne Universit´e, Universit´e Paris Diderot, CNRS, Institut de Math´ematiquesde Jussieu-Paris Rive Gauche, F-75005, Paris, FranceValentin Ovsienko, Centre national de la recherche scientifique, Laboratoire de Math´ematiques U.F.R.Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 Reims cedex 2, France
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