Linear Q-difference Equations Research Articles

A high-order q-difference equation for q-Hahn multiple orthogonal polynomials

A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case.

Lax forms of the q-Painlevé equations

All q-Painlevé equations which are obtained from the q-analog of the sixth Painlevé equation are expressed in a Lax formalism. They are characterized by the data of the associated linear q-difference equations. The degeneration pattern of the q-Painlevé equations is also presented.

DIFFERENCE AND DIFFERENTIAL EQUATIONS FOR THE COLORED JONES FUNCTION

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.

ARITHMETICAL RESULTS ON CERTAIN q-SERIES, I

Entire transcendental solutions of certain mth order linear q-difference equations with polynomial coefficients are considered. The aim of this paper is to give, under appropriate arithmetical conditions, lower bounds for the dimension of the K-vector space generated by 1 and the values of these solutions at m successive powers of q, where K is the rational or an imaginary quadratic number field. The main ingredients of the proofs are, first, Nesterenko's dimension estimate and its various generalizations, and secondly, Popov's method (in Töpfer's version) for the asymptotic evaluation of certain complex integrals.

Galois theory of q-difference equations

Choose q∈ℂ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. A systematic treatment of classification and moduli is developed. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M) on the Tate curve E q :=ℂ * /q ℤ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification ([At]) of the indecomposable vector bundles on the complex Tate curve. Linear q-difference equations are also studied in positive characteristic p in order to derive Atiyah’s results for elliptic curves for which the j-invariant is not algebraic over 𝔽 p .

Q-Dominant and q-recessive matrix solutions for linear quantum systems

In this study, linear second-order matrix q-difference equations are shown to be formally self-adjoint equations with respect to a certain inner prod- uct and the associated self-adjoint boundary conditions. A generalized Wronskian is introduced and a Lagrange identity and Abel's formula are established. Two reduction-of-order theorems are given. The analysis and characterization of q- dominant and q-recessive solutions at infinity are presented, emphasizing the case when the quantum system is disconjugate.

The General Theory of Linear Difference Equations over the Max-Plus Semi-Ring

We present the mathematical theory underlying systems of linear difference equations over the max-plus semi-ring. The result provides an analog of isomonodromy theory for ultradiscrete Painleve equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q-difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric PIV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.

TheC–polynomial of a knot

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U_q(SL_2)) specializes at q=1 to the better known A-polynomial of a knot, which has to do with genuine SL_2(C) representations of the knot complement. Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C-polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C-polynomial and the Alexander polynomial, all confirmed for the class of twist knots.

Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences

We describe q-analogs of asymptotic expansions and of multisummable series and we apply them to the local analytic classification of irregular linear q-difference equations, thus answering questions of G.D. Birkhoff. To cite this article: J.-P. Ramis et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

A q-Analog of the Garnier System

A q-difference analog of the Garnier system is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit is also studied.

Discreteq-derivatives and symmetries ofq-difference equations

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation

La filtration canonique par les pentes d'un module aux $q$-différences et le gradué associé

We show that the Newton polygon of a linear q-difference equation only depends on the corresponding q-difference module. We interpret the classical results of convergent factorisation of Adams-Birkhoff-Guenther in terms of the existence of a canonical filtration. Moreover, the associated graded module has excellent functorial and tensorial properties.

MEROMORPHIC SOLUTIONS OF q-DIFFERENCE EQUATIONS

Consider a linear q-difference equation: a(z)f(z) = b(z)f(qz) + c(z), 0 q| a(z), b(z), c(z) ∈ C[z]. We suppose that it admits a transcendental meromorphic solution f (z). Here we will determine the counting function N(r, f), and investigate the deficiency property of f (z). Finally, we give some examples.

AN INTRODUCTION TO THE H&lt;sub&gt;q&lt;/sub&gt;-SEMICLASSICAL ORTHOGONAL POLYNOMIALS

Orthogonal polynomials associated with Hq- semiclassical linear form will be studied as a generalization of the Hq-classical linear forms. The concept of class and a criterion for determining it will be given. The q-difference equation that the corresponding formal Stieltjes series satisfies is obtained. Also, the structure relation as well as the second order linear q-difference equation are obtained. Some examples of Hq-semiclassical of class 1 were highlighted.

Construction of recurrences for the coefficients of expansions in q-classical orthogonal polynomials

An algorithmic approach is given to construct recurrence relations for the coefficients of the series expansion of a solution of a linear q-difference equation in q-classical orthogonal polynomials (in the Hahn's sense).

A direct algorithm to compute rational solutions of first order linear [formula omitted]-difference systems

We present an algorithm to compute rational function solutions to a first order system of linear q-difference equations with rational coefficients. We make use of the fact that q-difference equations bear similarity to differential equations at the point 0 and to difference equations at other points. This allows the combining of known algorithms for the differential and the difference cases. This algorithm does not require preliminary uncoupling of the given system.

Partie irrégulière des solutions pour une équation linéaire aux [formula omitted]-différences

The goal of this Note is to give a new method of resolution for a linear q-differential equation in a neighborhood of the irregular singularity 0, different from the one Marotte and Zhang used in [3]. We will explicitly determined the mostly irregular part for a formal solution of a linear q-difference equation. To cite this article: A. Essadiq, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 139–144.

An algorithm for computing a standard form for second-order linear q-difference equations

In this article an algorithm is presented for computing a standard form for second order linear q-difference equations. This standard form is useful for determining the q-difference Galois group and the set of Liouvillian solutions of a given equation.

A q-analog of the sixth Painlev� equation

A q-difference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy-preserving deformation of linear differential equations. The continuous limit and special solutions in terms of q-hypergeometric functions are also discussed.

Quantum symmetries of q-difference equations

A general method is presented to determine the symmetry operators of linear q-difference equations. It is applied to q-difference analogs of the Helmoltz, heat, and wave equations in diverse dimensions. The symmetries are found to generate q-deformations of classical Lie algebras. The method of separation of variables is used to obtain solutions which are seen to involve many basic special functions. This allows the derivation of various identities and formulas for these q-functions.