On radius of convergence of q-deformed real numbers
Ludivine Leclere, Sophie Morier-Genoud, Valentin Ovsienko, Alexander Veselov
OON RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS LUDIVINE LECLERE, SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND ALEXANDER VESELOV
Abstract.
We study analytic properties of “ q -deformed real numbers”, a notion recently introducedby two of us. A q -deformed positive real number is a power series with integer coefficients in one formalvariable q . We study the radius of convergence of these power series assuming that q ∈ C . Our mainconjecture, which can be viewed as a q -analogue of Hurwitz’s Irrational Number Theorem, provides alower bound for these radii, given by the radius of convergence of the q -deformed golden ratio. Theconjecture is proved in several particular cases and confirmed by a number of computer experiments.For an interesting sequence of “Pell polynomials”, we obtain stronger bounds. Introduction
There is a famous result due to Hurwitz [6], which roughly claims that the golden ratio ϕ = √ isthe most irrational number. More precisely, for any real number x ∈ R one can define as the measure ofirrationality the Markov constant µ ( x ), which is the infimum of c , for which the inequality | x − pq | < cq holds for infinitely many integer p, q. Hurwitz’s Irrational Number Theorem claims that for every x ∈ R (1.1) µ ( x ) ≤ µ ( ϕ ) = 1 √ x which are PSL(2 , Z )-equivalent to the golden ratio.In this paper we discuss a possible q -analogue of this classical result. Namely, we consider the q -deformations (or “ q -analogues”) of real numbers, which have been recently introduced in [12, 13] andstudied further in [10, 14, 8]. They have several nice properties and connections, including theory ofConway-Coxeter friezes and knot invariants.For a rational x = rs > q -deformation is a rational function(1.2) (cid:104) rs (cid:105) q = R ( q ) S ( q ) , where R ( q ) and S ( q ) are polynomials with positive integer coefficients both depending on r and s . When x ≥ q -deformation of x is defined as a power series in q :(1.3) [ x ] q = 1 + κ q + κ q + κ q + · · · with coefficients κ k ∈ Z . To obtain the series (1.3), one chooses an arbitrary sequence of rationals ( x i ) i ∈ Z converging to x . It turns out that the Taylor series of the rational functions [ x i ] q stabilize, as i grows;see [13]. Moreover, the stabilized series depends only on x (and not on the approximating sequence ofrationals). The series [ x ] q is defined as the stabilization of the Taylor series of [ x i ] q . It is unknown, ingeneral, how to characterize the class of power series that represent q -deformed real numbers. One ofthe goals of this paper is to show that series arising in this context are not arbitrary. In particular, theymust have non-zero radius of convergence.We study analytic properties of q -deformed real numbers. Considering the parameter of deformation q as a complex variable , q ∈ C , we study the radius of convergence of the series (1.3).The following conjecture can be considered as a possible q -analogue of the Hurwitz claim. Conjecture 1.1.
For every real x ≥ , the radius of convergence R ( x ) of the series [ x ] q satisfies (1.4) R ( x ) ≥ R ∗ := R ( ϕ ) = 3 − √ and the equality holding only for x which are PSL(2 , Z ) -equivalent to ϕ = √ . a r X i v : . [ m a t h . QA ] F e b L. LECLERE, S. MORIER-GENOUD, V. OVSIENKO, AND A.VESELOV
Recall that the inverse ρ = R − of the radius of convergence of power series (1.3) can be given by thestandard formula (see e.g. [3])(1.5) ρ ( x ) = lim sup n →∞ | κ n | n and thus describes the growth of the coefficients κ n . Conjecture 1.1 can be reformulated in the form,more similar to Hurwitz, as the inequality(1.6) ρ ( x ) ≤ ρ ( ϕ ) = 3 + √ x , and thus means that [ ϕ ] q has the fastest growth of the coefficients among all q -deformed real numbers. In other words, the series [ x ] q always “converges better” than [ ϕ ] q , whenever x is not equivalent to ϕ . In the particular case where x = rs is rational, Conjecture 1.1 means that thepolynomial S ( q ) in (1.2) has no roots q with | q | < R ∗ ≈ . . We should mention that the analogy with the Diophantine analysis here cannot be extended to thecelebrated Markov theorem [9], claiming that the set of all possible Markov constants µ = µ ( x ) > isdiscrete (see e.g. [1]). The situation might be more similar to the Lyapunov spectrum of Markov andEuclid trees, which fills the whole segment [0 , ln ϕ ], see [16]. In particular, for every R such that R ∗ 1, there exists a number x , such that R ( x ) ≤ R .Conjecture 1.1 was checked by a long series of computer experimentation, and will be proved in severalparticular cases. Provided it is correct, this gives a restriction for the series (1.3) that can appear as a q -deformed real number. However, it is quite clear that the radius of convergence is not the only condition.It would be interesting to obtain more information about this class of power series.Let us give here an idea about the approach we use to prove the main statements. Our main analyticaltool is the classical Rouch´e theorem (see [3, 17]) that states the following. Let f and g be two functionsof one complex variable q , holomorphic inside a disc D and continuous on the bound ∂D . Suppose that f ( q ) > g ( q ) on ∂D , then f and f + g have the same number of zeros inside D . We will usually supposethat f has no zeros inside D , and conclude that the same is true for f + g .The paper is organized as follows.In Secion 2, we briefly recall the notion of q -rationals and q -irrationals. Following [8], we emphasizethe role of the modular group PSL(2 , Z ). We also give explicit formulas in terms of continued fractions.In Section 3, we consider the important example of q -deformed golden ratio. Its approximation by thequotients of consecutive Fibonacci numbers leads to interesting “Fibonacci polynomials”. We check thestatement of Conjecture 1.1 in this particular case.In Section 4, we consider the q -number (cid:2) √ (cid:3) q , and its approximation by the quotients of consecutive“Pell polynomials”. We strengthen the statement of Conjecture 1.1 in this case, replacing R ∗ by another,greater value.Finally, in Section 5 we prove the statement of Conjecture 1.1 for a large class of irrational numberswith a certain restriction (that, as we believe, is technical and can perhaps be removed). We give twomore interesting examples.2. Definition and explicit formulas for q -reals We start with recalling an axiomatic definition of q -deformed rational numbers. For more equivalentdefinitions; see [12, 14].2.1. An axiomatic definition. Recall that every rational can be obtained from 0, by applying a se-quence of the operations x (cid:55)→ x + 1 and x (cid:55)→ − x .The following two recurrences (see [8], and also [14]) suffice for calculation of q -rationals. Definition 2.1. The q -deformation sends every rational number x = rs to a rational function in qx (cid:55)−→ [ x ] q = R ( q ) S ( q ) , N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 3 in such a way that the following two recurrent formulas are satisfied(2.1) [ x + 1] q = q [ x ] q + 1 , (cid:20) − x (cid:21) q = − q [ x ] q . Recurrences (2.1) determine the q -deformation of a rational in a unique way, starting from the “initial”trivial deformation [0] q = 0. Example 2.2. The following examples are easy to obtain applying (2.1).(a) The q -deformation of integers corresponds to the standard formulas of Euler and Gauss. For n ∈ N ,one has [ n ] q = 1 + q + q + · · · + q n − , [ − n ] q = − q − n − q − n − q − n − · · · − q − . Both cases can be written as [ x ] q = − q x − x , where x is integer.(b) Already the simplest example of shows that the q -deformation is not the quotient of the q -deformed integers appearing in numerator and denominator: (cid:20) (cid:21) q = q q , (cid:20) − (cid:21) q = − q (1 + q ) . The above expression [ x ] q = − q x − x is no longer true.(c) The next examples (cid:20) (cid:21) q = 1 + 2 q + q + q q , (cid:20) (cid:21) q = 1 + q + 2 q + q q + q illustrate the fact that the numerator and denominator in (1.2) depend simultaneously on r and s . Indeed,the “quantized 5” in the numerator depends on the denominator. Remark 2.3. In the case of q -rationals, one also has the formula for the negation and inverse:(2.2) [ − x ] q = − [ x ] q − q , (cid:20) x (cid:21) q = 1[ x ] q − . The action and a central extension of PSL(2 , Z ) . We briefly mention here a more conceptualway of understanding Definition 2.1. This observation will not be used in the sequel, and will be part ofa separate work.Recurrences (2.1) can be reformulated as a statement that “ q -deformation commutes with PSL(2 , Z )-action”. Indeed, recurrences (2.1) define an action of the modular group PSL(2 , Z ) on q -deformed ratio-nals. This action is given by fractional-linear transformations and is generated by the matrices(2.3) T q = (cid:32) q 10 1 (cid:33) , S q = (cid:32) − q (cid:33) which are q -deformations of the standard generators, T, S , of PSL(2 , Z ) corresponding to q = 1 in (2.3).The relations S = Id and ( T S ) = Id, defining PSL(2 , Z ), become S q = q Id , ( T q S q ) = q Id , The matrices T q , S q generate an interesting non-trivial central extension of PSL(2 , Z ), which is differentfrom the braid group B , yet contains it as a subgroup. The centre, { q n Id | n ∈ Z } , of the extended mod-ular group acts trivially of q -deformed rationals, so that one still has the well-defined action of PSL(2 , Z ). Remark 2.4. (a) Let us stress on the fact that emergence of a central extension of the symmetry groupis a usual situation in geometry; see [7]. However, the centre of the extended group usually acts triviallyon quantized objects.(b) The matrix S q arose in the context of quantum groups; see [2], while T q can be viewed as a“standard” matrix connected to quantum integers. However, we did not find in the literature simultaneousappearance of T q and S q . L. LECLERE, S. MORIER-GENOUD, V. OVSIENKO, AND A.VESELOV q -deformed continued fractions. Every rational number rs > r, s ∈ Z > are coprime,has a standard finite continued fraction expansion rs = a + 1 a + 1 . . . + 1 a m , where a i ≥ a ≥ rs = [ a , . . . , a m ]. Note that, choosing aneven number of coefficients, one removes the ambiguity [ a , . . . , a n , 1] = [ a , . . . , a n + 1] and makes theexpansion unique.There is also a unique continued fraction expansion with minus signs, often called the Hirzebruch-Jungcontinued fraction: rs = c − c − . . . − c k , where c j ≥ c ≥ rs = (cid:74) c , . . . , c k (cid:75) . The coefficients a i and c j of the above expansions are connected by the Hirzebruch formula; see, e.g., [5, 11] and Section 5.1. Definition 2.5. (a) The q -deformed regular continued fraction is defined by(2.4) [ 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 [ a ] q is the Euler q -integer.(b) The q -deformed Hirzebruch-Jung continued fraction is(2.5) (cid:74) c , . . . , c k (cid:75) q := [ c ] q − q c − [ c ] q − q c − . . . . . . [ c k − ] q − q c k − − [ c k ] q For every rational rs written in two different ways: rs = [ a , . . . , a m ] = (cid:74) c , . . . , c k (cid:75) , the rational functions (2.4) and (2.5) coincide, and also coincide with the rational function (cid:2) rs (cid:3) q providedby Definition 2.1; see [12].2.4. Stabilization phenomenon, q -irrationals. The notion of q -deformed rational was extended toirrational numbers in [13]. Let x ≥ x n ) n ≥ ,converging to x . Consider the corresponding sequence of q -rationals [ x ] q , [ x ] q , . . . It turns out that thissequence of rational functions also converge, but in the sense of formal power series. N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 5 Consider the Taylor expansions at q = 0 of the rational functions [ x n ] q , that, abusing notation, wealso denote by [ x n ] q : [ x n ] q = (cid:88) k ≥ κ n,k q k . One has the following stabilization property. Theorem 2.1 ([13]) . (i) For every k ≥ , the coefficients κ n,k of the Taylor series of the functions [ x n ] q stabilize, as n grows.(ii) The limit coefficients, κ k := lim n →∞ κ n,k , do not depend on the choice of the sequence of rationals,but only on the irrational number x . The q -deformation, [ x ] q , is the limit power series in q , which has the form (1.3) (i.e., the coefficient ofdegree 0 is equal to 1). Furthermore, the recurrence (2.1) also holds for the series [ x ] q , and allows one toextend the q -deformation to the case of x < 0. The resulting series in q is a Laurent series (with integercoefficients): [ x ] q = − q − N + κ − N q − N + κ − N q − N + · · · where N ∈ Z > such that − N ≤ x < − N . Remark 2.6. Let us mention that the stabilization phenomenon fails when a sequence of rationals ( x n ) n ≥ ,converges to another rational, cf. [13].3. Convergence radius of q -golden ratio and roots of the Fibonacci polynomials The simplest example of q -irrational is the q -deformation of the celebrated golden ratio, ϕ = √ . Theseries [ ϕ ] q is obtained as stabilized Taylor series of the q -deformed quotients of the consecutive Fibonaccinumbers (cid:104) F n +1 F n (cid:105) q (see [12, 13]). We call the polynomials in the numerator and the denominator of theserational functions the Fibonacci polynomials .We prove that the radius of convergence of [ ϕ ] q is R ∗ = −√ , and that all roots of the Fibonaccipolynomials belong to the annulus bounded by the circles with the radius R ∗ = −√ and R − ∗ = √ .This result is in an accordance with Conjecture 1.1.3.1. The q -deformed golden ratio. The q -deformation of the golden ratio, ϕ = √ , was consideredin [13]. The series [ ϕ ] q can be written as an infinite continued fraction:(3.1) [ ϕ ] q = 1 + q q + 11 + q q + 1 . . . = 1 + 1 q − + 1 q + 1 q − + 1 . . . The series 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 · · · The sequence of coefficients in [ ϕ ] q coincides (up to the alternating sign) with the remarkable sequenceA004148 of [15] called the “generalized Catalan numbers”.The series [ ϕ ] q is a solution of the functional equation(3.2) q X = (cid:0) q + q − (cid:1) X + 1 , which can be deduced from (3.1). Proposition 3.1. The radius of convergence of the series [ ϕ ] q is equal to R ∗ . L. LECLERE, S. MORIER-GENOUD, V. OVSIENKO, AND A.VESELOV Proof. It follows from (5.2), that the generating function of [ ϕ ] q can therefore be written in radicals:(3.3) GF [ ϕ ] q = q + q − (cid:112) ( q + 3 q + 1)( q − q + 1)2 q , and the series [ ϕ ] q is the Taylor expansion of GF [ ϕ ] q at q = 0. The number R ∗ is the modulus of thesmallest (i.e., closest to 0) root of the polynomials under the radical in (3.3). Indeed, q + 3 q + 1 = ( q + R ∗ ) (cid:0) q + R − ∗ (cid:1) , and therefore the Taylor series of (3.3) converges for | q | < R ∗ . (cid:3) Remark 3.2. Observe that formula (3.1) has a certain similarity with the celebrated Rogers-Ramanujancontinued fraction R ( q ) = 1 + 11 + q q q . . . but the deformation (3.1) has very different properties.3.2. The Fibonacci polynomials. The most natural choice of a sequence of rationals converging to ϕ is related to a quite remarkable and well-known sequence of polynomials.Let F n be the n th Fibonacci number, the sequence of rationals F n +1 F n converges to ϕ . Quantizing thissequence, one obtains a sequence of rational functions(3.4) (cid:20) F n +1 F n (cid:21) q =: ˜ F n +1 ( q ) F n ( q ) . The polynomials ˜ F n +1 ( q ) and F n ( q ) in the numerator and denominator of (3.4) are q -deformations ofthe Fibonacci numbers, considered in [12].Both sequences of polynomials F n ( q ) and ˜ F n ( q ) are of degree n − n ≥ 2) and are mirror of eachother: ˜ F n ( q ) = q n − F n ( q − ) . The polynomials F n ( q ) can be calculated recursively. It will be convenient to separate the sequenceof polynomials F n ( q ) into two subsequences, with even n and odd n . Both of these sequences satisfy thesame recurrence, which is a q -analogue of the classical recurrence F n +2 = 3 F n − F n − for the Fibonacci numbers. We have the following. Proposition 3.3. The polynomials F n ( q ) in the denominator of (3.4) are determined by the recurrence (3.5) F n +2 ( q ) = [3] q F n ( q ) − q F n − ( q ) , where [3] q = 1 + q + q , and the initial conditions ( F ( q ) = 0 , F ( q ) = 1) and ( F ( q ) = 1 , F ( q ) = 1 + q ) . Proof. It was shown in [12] that F (cid:96) +1 = q F (cid:96) + F (cid:96) − , F (cid:96) +2 = F (cid:96) +1 + q F (cid:96) , Recurrence (3.5) follows readily. (cid:3) N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 7 The coefficients of the polynomials F n ( q ) and ˜ F n ( q ) form the triangles11 11 1 11 2 1 11 2 2 2 11 3 3 3 2 11 3 4 5 4 3 1 · · · 11 11 1 11 1 2 11 2 2 2 11 2 3 3 3 11 3 4 5 4 3 1 · · · known as Sequences A123245 and A079487 of OEIS [15], respectively. Example 3.4. One has (cid:2) (cid:3) q = 1 + q + 2 q + q q + q , (cid:2) (cid:3) q = 1 + 2 q + 2 q + 2 q + q q + q + q , (cid:2) (cid:3) q = 1 + 2 q + 3 q + 3 q + 3 q + q q + 2 q + 2 q + q , (cid:2) (cid:3) q = 1 + 3 q + 4 q + 5 q + 4 q + 3 q + q q + 3 q + 3 q + 2 q + q ,. . . . . . . . . Remark 3.5. The polynomials F n ( q ) with odd n are specializations of 3-parameter family of polynomialsconsidered in [4] (see Remark 8.4.).3.3. Roots of the Fibonacci polynomials. Our next goal is to obtain the bounds for the absolutevalues of roots of the Fibonacci polynomials.Let us use the following notation. Let D be the disc and C the circle with radius R ∗ : D = { q ∈ C , | q | ≤ R ∗ } , C = { q ∈ C , | q | = R ∗ } . Theorem 3.1. For every n ∈ N , and for every root q r , of each of the polynomials, F n ( q ) and ˜ F n ( q ) ,one has R ∗ < | q r | < R − ∗ . Proof. Recurrence (3.5) can be rewritten as follows(3.6) F n +2 ( q ) F n ( q ) = [3] q − q F n − ( q ) F n ( q ) . By the Rouch´e theorem (see, e.g., [3, 17] and the introduction), it suffices to prove that, for every n ,(3.7) (cid:12)(cid:12) q + q + 1 (cid:12)(cid:12) C > R ∗ (cid:12)(cid:12)(cid:12)(cid:12) F n − ( q ) F n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C , on the circle C , in order to prove, that, for every n , the polynomial F n ( q ) has no roots inside the disc D with radius R ∗ . Indeed, the cyclotomic polynomial [3] q = q + q + 1 has no roots inside D , so that (4.6)will guarantee, by induction on n , that F n +2 ( q ) has no roots inside D .The inequality (4.6) follows from the next two lemmas. Lemma 3.6. On the circle C , one has R ∗ ≤ (cid:12)(cid:12) q + q + 1 (cid:12)(cid:12) C ≤ R ∗ , where R ∗ is the minimal, and R ∗ the maximal value of [3] q on the circle. Lemma 3.7. Assume that (cid:12)(cid:12)(cid:12) F n − ( q ) F n ( q ) (cid:12)(cid:12)(cid:12) C < R ∗ , then (cid:12)(cid:12)(cid:12) F n ( q ) F n +2 ( q ) (cid:12)(cid:12)(cid:12) C < R ∗ . L. LECLERE, S. MORIER-GENOUD, V. OVSIENKO, AND A.VESELOV Lemma 3.6 is a standard exercise of complex analysis. The minimum and maximum values of thefunction (cid:12)(cid:12) q + q + 1 (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( q + e iπ/ )( q + e − iπ/ ) (cid:12)(cid:12)(cid:12) on the circle C are attained at the points q = − R ∗ and q = R ∗ , respectively. Therefore, we have R ∗ − R ∗ + 1 ≤ (cid:12)(cid:12) q + q + 1 (cid:12)(cid:12) C ≤ R ∗ + R ∗ + 1 . Lemma 3.6 then follows from the relation R ∗ = 3 R ∗ − R ∗ = −√ . (cid:3) Lemma 3.7 follows directly from (3.6) and Lemma 3.6. (cid:3) It remains to prove that the polynomials ˜ F n ( q ) have no roots inside D . The proof is similar,since ˜ F n ( q ) also satisfy (3.5). This follows from the recurrent formulas˜ F (cid:96) +1 = ˜ F (cid:96) + q ˜ F (cid:96) − , ˜ F (cid:96) +2 = q ˜ F (cid:96) +1 + ˜ F (cid:96) , proved in [12]. Theorem 3.1 is proved. (cid:3) It is easy to prove that the bounds R ∗ and R − ∗ of Theorem 3.1 are tight. This follows from the factthat R ∗ is the radius of convergence of [ ϕ ] q , and the stabilization phenomenon.4. Radius of (cid:2) √ (cid:3) q and roots of the Pell polynomials The second example we consider is √ 2. It is important in Markov theory of Diophantine approxima-tions. The slight modification, √ √ Pell numbers P n +1 P n = [2 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n .We show that the convergence radius of the series (cid:2) √ (cid:3) q and (cid:2) √ (cid:3) q is equal to R √ := R ( √ R √ = 1 + √ − (cid:112) √ − ≈ . , and that the roots of the polynomials in the numerators and denominators of (cid:104) P n +1 P n (cid:105) q belong to theannulus bounded by the circles with radius R √ and R − √ .4.1. The series (cid:2) √ (cid:3) q and (cid:2) √ (cid:3) q . These series are related by (cid:2) √ (cid:3) q = q (cid:2) √ (cid:3) q + 1 (cf. (2.1))and obviously have the same convergence radius; we prefer to perform the calculations for (cid:2) √ (cid:3) q .The q -deformation (cid:2) √ (cid:3) q is given by the infinite 2-periodic continued fraction(4.2) (cid:104) √ (cid:105) q = 1 + q + q q + q + 11 + q + q q + q + 1 . . . see [13]. This is the q -deformed classical continued fraction expansion √ , , , , . . . ].The series (cid:2) √ (cid:3) q satisfies the following functional equation:(4.3) qX − (cid:0) q + 2 q − (cid:1) X − , readily obtained from (4.2), and can be calculated from it recursively: (cid:2) √ (cid:3) 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 · · · N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 9 see Sequence A337589 of [15] for the coefficients of this series. Proposition 4.1. The radius of convergence of the series (cid:2) √ (cid:3) q and (cid:2) √ (cid:3) q is equal to R √ .Proof. The generating function of the series can be deduced from (4.3):GF[ √ ] q = q + 2 q − (cid:112) ( q + q + 4 q + q + 1)( q − q + 1)2 q . The radius (4.1) is equal to the modulus of the root of the polynomial q + q + 4 q + q + 1 closest tozero. (cid:3) The Pell polynomials. The irrational (cid:2) √ (cid:3) q can be approximated by the quotient of theconsecutive Pell numbers: P n +1 P n = [2 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n . We define the Pell polynomials via (cid:20) P n +1 P n (cid:21) q =: ˜ P n +1 ( q ) P n ( q ) . The polynomials ˜ P n ( q ) and P n ( q ) are of degree 2 n − 3, and, similarly to the Fibonacci polynomials, arethe mirrors of each other: q n − ˜ P n ( q − ) = P n ( q ). Proposition 4.2. The polynomials P n ( q ) are determined by the recurrence (4.4) P n +2 = (cid:18) (cid:19) q P n − q P n − , where (cid:0) (cid:1) q = 1 + q + 2 q + q + q is the Gaussian q -binomial, and the initial conditions ( P ( q ) = 0 , P ( q ) = 1 + q ) and (cid:0) P ( q ) = 1 , P ( q ) = 1 + q + 2 q + q (cid:1) . Proof. Recurrence (4.4) follows from the formulas P (cid:96) +1 = (cid:0) q + q (cid:1) P (cid:96) + P (cid:96) − , P (cid:96) +2 = (1 + q ) P (cid:96) +1 + q P (cid:96) , proved in [12]. (cid:3) The coefficients of P n ( q ) form a triangular sequence11 11 1 2 11 2 3 3 2 11 2 5 6 6 5 3 11 3 7 11 13 13 11 7 3 11 3 9 16 24 29 29 25 18 10 4 1 · · · (see Sequence A323670 of [15]). Figure 1. Roots of P ( q ): the smallest root modulus is 0 . . . . , and the largest 1 . . . . Example 4.3. One has (cid:2) (cid:3) q = 1 + 2 q + q + q q + q , (cid:2) (cid:3) q = 1 + 2 q + 3 q + 3 q + 2 q + q q + 2 q + q , (cid:2) (cid:3) q = 1 + 3 q + 5 q + 6 q + 6 q + 5 q + 2 q + q q + 3 q + 3 q + 2 q + q , (cid:2) (cid:3) q = 1 + 3 q + 7 q + 11 q + 13 q + 13 q + 11 q + 7 q + 3 q + q q + 5 q + 6 q + 6 q + 5 q + 3 q + q ,. . . . . . . . . Roots of the Pell polynomials. We prove that the roots of the Pell polynomials always belongto the annulus bounded by R √ and R − √ ; see Figure 4.3. Theorem 4.1. The roots of the polynomials, P n ( q ) and ˜ P n ( q ) belong to the annulus bounded by R √ and R − √ : R √ < | q r | < R − √ , for every q r , such that P n ( q r ) = 0 , or ˜ P n ( q r ) = 0 .Proof. Recurrence (4.4) lads to(4.5) P n +2 ( q ) P n ( q ) = (cid:18) (cid:19) q − q P n − ( q ) P n ( q ) . Once again, we use the Rouch´e theorem and want to prove that, for every n , the absolute value of thepolynomial (cid:0) (cid:1) q satisfies(4.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C > R √ (cid:12)(cid:12)(cid:12)(cid:12) P n − ( q ) P n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C , on the circle C = (cid:8) | q | = R √ (cid:9) . Indeed, all the roots of the polynomial (cid:0) (cid:1) q = ( q + 1)( q + q + 1) belongto the unit circle, so that (4.6) will guarantee, by induction on n , that P n +2 ( q ) has no roots inside D .The inequality (4.6) follows from the next two lemmas. N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 11 Lemma 4.4. On the circle C , one has R √ (cid:16) R √ + R √ + 1 (cid:17) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C . Proof of Lemma 4.4 . The minimum of the function (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) ( q + 1)( q + q + 1) (cid:12)(cid:12) on the circle C is attained at the point q = − R √ . Therefore, we have( R √ + 1)( R √ − R √ + 1) = R √ − R √ + 2 R √ − R √ + 1 ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C . Since the radius R √ satisfies the equation R √ − R √ + R √ − R √ + 1 = 0 , the above inequalitybecomes R √ + R √ + R √ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C . Hence the lemma. (cid:3) Lemma 4.5. Assume that (cid:12)(cid:12)(cid:12) P n − ( q ) P n ( q ) (cid:12)(cid:12)(cid:12) C < R √ , then (cid:12)(cid:12)(cid:12) P n ( q ) P n +2 ( q ) (cid:12)(cid:12)(cid:12) C < R √ .Proof of Lemma 4.5 . It follows from (4.5) that (cid:12)(cid:12)(cid:12)(cid:12) P n +2 ( q ) P n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C − R √ (cid:12)(cid:12)(cid:12)(cid:12) P n − ( q ) P n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C . Lemma 4.4 and assumption (cid:12)(cid:12)(cid:12) P n − ( q ) P n ( q ) (cid:12)(cid:12)(cid:12) C < R √ then imply (cid:12)(cid:12)(cid:12)(cid:12) P n +2 ( q ) P n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C ≥ R √ + R √ . Hence the lemma. (cid:3) The proof that the polynomials ˜ P n ( q ) have no roots inside D is analogous since ˜ P n ( q ) also satisfy (4.4).Theorem 4.1 is proved. (cid:3) A general result In this section, we prove that Conjecture 1.1 holds for x ∈ R , satisfying under some technical restric-tions on the coefficients of the continued fraction.5.1. Statement of the theorem. Given an irrational x ≥ 0, consider the Hirzebruch-Jung continuedfraction expansion x = (cid:74) c , c , c , . . . (cid:75) , see Section 2.3. Theorem 5.1. If every coefficient of the Hirzebruch-Jung continued fraction expansion x = (cid:74) c , c , c , . . . (cid:75) of an irrational x > satisfies (5.1) c i ≥ , for all i ≥ N , and some fixed N , then the radius of convergence of [ x ] q is greater or equal to R ∗ = −√ . Let us reformulate the inequality (5.1) in terms of the coefficients of the regular continued fractionexpansion x = [ a , a , a . . . ]. Recall the formula [5] expressing the coefficients of the Hirzebruch-Jungcontinued fraction: x = (cid:74) a + 1 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) a − , a + 2 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) a − , a + 2 , . . . , a n − + 2 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) a n − , . . . (cid:75) In other words, the coefficients with odd indices, a n − become a n − + 2 (except for a that becomes a + 1), and coefficients with even indices, a n produce an ( a n − (cid:40) a M − ≥ ,a M = 1 , starting from some M .5.2. Proof of Theorem 5.1. Using the PSL(2 , Z )-invariance of the q -deformation, we can assume thatcondition (5.1) is satisfied by the coefficients c i , for all i ≥ x by the finite convergents of the Hirzebruch-Jung con-tinued fraction: r n s n = (cid:74) c , c , c , . . . , c n (cid:75) , and consider the q -deformation R n ( q ) S n ( q ) := (cid:20) r n s n (cid:21) q . It suffices to prove that, for every n , the polynomial S n ( q ) has no roots in the disc D .Formula (2.5) implies that the polynomial S n ( q ) satisfy the recurrence S n +1 ( q ) = [ c n +1 ] q S n ( q ) − q c n − S n − ( q ) , with the initial values S ( q ) = 0 and S ( q ) = 1, that we rewrite as follows(5.2) S n +1 ( q ) S n ( q ) = [ c n +1 ] q − q c n − S n − ( q ) S n ( q ) . By the Rouch´e theorem, we need to prove that, for every n , the polynomial [ c n +1 ] q = − q cn +1 − q dominatesthe second summand of the right-hand-side of (5.2), when restricted on the circle C :(5.3) (cid:12)(cid:12)(cid:12) [ c n +1 ] q (cid:12)(cid:12)(cid:12) C > R c n − ∗ (cid:12)(cid:12)(cid:12)(cid:12) S n − ( q ) S n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C . Since for every positive integer c the polynomial [ c ] q has no roots in D , we will then argue by inductionthat S n +1 ( q ) also has no roots in D .For every positive integer c , one obviously has(5.4) 1 − R c R ≤ (cid:12)(cid:12)(cid:12) [ c ] q (cid:12)(cid:12)(cid:12) , on any circle of radius R .We will assume, by induction, that (cid:12)(cid:12)(cid:12) S n − ( q ) S n ( q ) (cid:12)(cid:12)(cid:12) C < R ∗ , and therefore R c n − ∗ (cid:12)(cid:12)(cid:12)(cid:12) S n − ( q ) S n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C < R c n − ∗ . We then have from (5.2) and (5.4) that (cid:12)(cid:12)(cid:12)(cid:12) S n +1 ( q ) S n ( q ) (cid:12)(cid:12)(cid:12)(cid:12) C > − R c n +1 ∗ R ∗ − R c n − ∗ . To prove the induction step, we need the following. Lemma 5.1. If c i ≥ for all i , then (cid:12)(cid:12)(cid:12) S n +1 ( q ) S n ( q ) (cid:12)(cid:12)(cid:12) C > R ∗ .Proof. Indeed, 1 − R c n +1 ∗ R ∗ − R c n − ∗ − R ∗ = 1 − R c n +1 ∗ − R c n − ∗ − R c n − ∗ − R ∗ − R ∗ R ∗ . Since c i ≥ i , we have1 − R c n +1 ∗ − R c n − ∗ − R c n − ∗ − R ∗ − R ∗ R ∗ ≥ − R ∗ − R ∗ − R ∗ − R ∗ R ∗ = 14 − R ∗ R ∗ , N RADIUS OF CONVERGENCE OF q -DEFORMED REAL NUMBERS 13 because R ∗ = 3 R ∗ − 1. One checks that 7 − R ∗ > (cid:3) We proved that (5.3) holds for all n . Theorem 5.1 is proved. Remark 5.2. Let us stress on the fact that, using the adopted approach, we are unable to improve theassumption (5.1). Indeed, assuming c i ≥ − R ∗ ,which is negative. However, the statement of the theorem is true in the case c i = 3 for all i ≥ 1, thatcorresponds to the Fibonacci polynomials, see Section 3.5.3. Miscellaneous experiments. Multiple computer experiments show that the situation in the q -deformed case is very different from classical Markov theory [1]. Example 5.3. For √ 3, one has: √ , , 2] = (cid:74) , (cid:75) . The generating function of the series (cid:2) √ (cid:3) q isGF[ √ ] q = q + q − q − (cid:112) q + 2 q + 3 q + 3 q + 2 q + 12 q , see [13]. The absolute value of the minimal root of the polynomial under the radical is R √ := R ( √ ≈ . . . . which is between R ∗ and R √ : R ∗ < R √ < R √ . This example demonstrates, that, unlike classical Markov theory [1], the series corresponding to √ √ ϕ and the ”silver ratio” √ Example 5.4. The number α = √ = [2 , , , (cid:104) √ (cid:105) q is R bronze := R ( α ) = 1 + √ − (cid:113) (cid:0) √ − (cid:1) ≈ . . . . Indeed, a direct computation gives (cid:104) √ (cid:105) q = q +2 q +3 q +3 q + q − √ ( q +3 q +5 q +3 q +1)( q +2 q +3 q +5 q +3 q +2 q +1)( q − q +1)2 q ( q +2 q + q +1) Note that 221 = 13 · 17, the factors under the radical are q -versions of these numbers. The radius ofconvergence is equal to the absolute value of the minimal root of the polynomial under the radical, whichcan be found explicitly.We wonder if for other quadratic irrationals the radius of convergence is an algebraic number ofdegree 2 n (like in the theory of ruler-and-compass construction) as it was for ϕ, √ 2, and for the aboveexample, but we have no explicit formulas in general. Acknowledgements . We are very grateful to the Mathematisches Forschungsinstitut Oberwolfach forthe hospitality during our RiP stay in summer 2020, when this project was started. We are pleasedto thank Jenya Ferapontov and Sergei Tabachnikov for fruitful discussions. This paper was partiallysupported by the ANR project ANR-19-CE40-0021. References [1] M. Aigner, Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from IrrationalNumbers to Perfect Matchings . Springer, 2013.[2] J. Bernstein, T. Khovanova, On the quantum group SL q (2), Comm. Math. 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Oxford University Press, Oxford, 1958. x+454 pp. Ludivine Leclere, Sophie Morier-Genoud, Sorbonne Universit´e, Universit´e Paris Diderot, CNRS, Institutde Math´ematiques de Jussieu-Paris Rive Gauche, F-75005, Paris, FranceValentin Ovsienko, Centre National de la Recherche Scientifique, Laboratoire de Math´ematiques, Univer-sit´e de Reims, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 Reims cedex 2,FranceAlexander Veselov, Department of Mathematical Sciences, Loughborough University, Loughborough LE113TU, UK, Moscow State University and Steklov Mathematical Institute, Moscow, Russia Email address ::