Order Q-difference Equations Research Articles

Zero order meromorphic solutions of q-difference equations of Malmquist type

We consider the first order q-difference equation(†)f(qz)n=R(z,f), where q≠0,1 is a constant and R(z,f) is rational in both arguments. When |q|≠1, we show that, if (†) has a zero order transcendental meromorphic solution, then (†) reduces to a q-difference linear or Riccati equation, or to an equation that can be transformed to a q-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of (†) are presented. Given that (†) can be transformed into a difference equation, we proceed to discuss the growth of the composite function f(ω(z)), where ω(z) is an entire function satisfying ω(z+1)=qω(z), and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for q-difference equations.

Holomorphic solutions of some functional equations related to nonlinear second order q-difference equations which has only one characteristic value

Here we consider the following functional equation, where and are holomorphic functions in , , , and x is a function of t such that . If we have a solution of nonlinear second order difference equations, then we can write general solutions by making use of a solution Ψ. In this present paper, we will prove the existence of a solution of the above functional equation, after we will show a relationship between the above functional equation and nonlinear second order q-difference equations which has only one characteristic value.

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

The Factorization Method for the β-Difference Hypergeometric Equation

We discuss special types of the factorization method for the so-called β-difference hypergeometric equation, and then illustrate them by solving or transforming difference and q-difference hypergeometric equations, including the Askey-Wilson second order q-difference equation.

A class of symmetric q-orthogonal polynomials with four free parameters

By using a generalization of Sturm–Liouville problems in q-difference spaces, a class of symmetric q-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order q-difference equation, the explicit form of the polynomials in terms of basic hypergeometric series, a three term recurrence relation and a general orthogonality relation are presented. Some particular examples are then studied in detail.

A few remarks on Poincaré-Perron solutions and regularly varying solutions

AbstractWe establish conditions guaranteeing the existence of (generalized) regularly varying solutions tonth order linear differential andq-difference equations. The proofs are based mainly on classical Poincaré’s and Perron’s theorems and certain transformations. In some special cases, our results reduce to the existing ones, thus actually we offer an alternative approach to some parts of the asymptotic theory of differential equations made in the framework of regular variation. For higher order cases, our statements are essentially new. Another important feature of this paper is that it reveals connections among various results and somehow revises some of them.

Using [formula omitted]-operators to construct orthogonal polynomials satisfying higher order q-difference equations

Let (pn)n be either the q-Meixner or the q-Laguerre polynomials. We form a new sequence of polynomials (qn)n by considering a linear combination of two consecutive pn: qn=pn+βnpn−1, βn∈R. Using the concept of D-operator, we generate sequences (βn)n for which the polynomials (qn)n are orthogonal with respect to a measure and common eigenfunctions of a higher order q-difference operator.

Asymptotic formulae for eigenvalues and eigenfunctions of q -Sturm-Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q-Sturm-Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q-trigonometric functions as well as fundamental sets of solutions of the associated second order q-difference equation. As in classical Sturm-Liouville theory, the eigenvalues behave like zeros of q-trigonometric functions and the eigenfunctions behave like q-trigonometric functions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Q-Karamata functions and second order q-difference equations

In this paper we introduce and study q-rapidly varying functions on the lattice q N0 := {q k : k ∈ N0}, q > 1, which naturally extend the recently established concept of q-regularly varying functions. These types of functions together form the class of the so-called q-Karamata functions. The theory of q-Karamata functions is then applied to half-linear q-difference equations to get information about asymptotic behavior of nonoscillatory solutions. The obtained results can be seen as q-versions of the existing ones in the linear and half-linear differential equation case. However two important aspects need to be emphasized. First, a new method of the proof is presented. This method is designed just for the q-calculus case and turns out to be an elegant and powerful tool also for the examination of the asymptotic behavior to many other q-difference equations, which then may serve to predict how their (trickily detectable) continuous counterparts look like. Second, our results show that q N0 is a very natural setting for the theory of q-rapidly and q-regularly varying functions and its applications, and reveal some interesting phenomena, which are not known from the continuous theory.

THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures ( q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.

A FINE DREAM

We shall develop further N. J. Fine's theory of three parameter non-homogeneous first order q-difference equations. The object of our work is to bring the Rogers–Ramanujan identities within the purview of such a theory. In addition, we provide a number of new identities.

Second order q-difference equations solvable by factorization method

By solving an infinite nonlinear system of q-difference equations one constructs a chain of q-difference operators. The eigenproblems for the chain are solved and some applications, including the one related to q-Hahn orthogonal polynomials, are discussed. It is shown that in the limit q → 1 the present method corresponds to the one developed by Infeld and Hull.

Linear Independence of the Values of q-Hypergeometric Series and Related Functions

We give linear independence results for the values of certain entire series and of functions satisfying certain first order q-difference equations. The former generalizes a result of Bezivin, while the latter refines that of the second named author in qualitative form. These results imply linear independence of the values of q-hypergeometric series.

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

Quantization scheme for modular q-difference equations

Modular pairs of some second order q-difference equations are considered. These equations may be interpreted as a quantum mechanics of a sort of hyperelliptic pendulum. It is shown the quantization of a spectrum may be provided by the condition of the analyticity of the wave function. Baxter's t-Q equations for the quantum relativistic Toda chain in the ``strong coupling regime'' are related to the system considered, and the quantization condition for Q-operator is also considered.

A Fourth Order $q$-Difference Equation for Associated Discrete $q$-Orthogonal Polynomials

In this work we prove that the associated polynomials of general q-orthogonal polynomials satisfy a fourth order q-difference equation. We provide two algorithms for constructing this equation and we identify its solution basis.

Deriving exact energy solutions to the symmetrized q-difference Harper equation.

This paper deals with the application of the q calculus to second order q-difference equations, like the q symmetrized form of the Harper equation. One obtains three-term recurrence relations, for which a symmetrized version is written down. This opens the way to establish explicit energy results that are dependent on the commensurability parameter. The continuous realization of such energy results exhibits interesting patterns characterized by hierarchical structures.

Fourth order q-difference equation for the first associated of the q-classical orthogonal polynomials

We derive the fourth-order q-difference equation satisfied by the first associated of the q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials σ and τ which appear in the q-Pearson difference equation D q ( σ ϱ) = τϱ defining the weight ϱ of the q-classical orthogonal polynomials inside the q-Hahn tableau.

COMPLEX q-ANALYSIS AND SCALAR FIELD THEORY ON A q-LATTICE

We develop the basic formalism of complex q-analysis to study the solutions of second order q-difference equations which reduce, in the q → 1 limit, to the ordinary Laplace equation in Euclidean and Minkowski space. After defining an inner product on the function space we construct and study the properties of the solutions, and then apply this formalism to the Schrödinger equation and two-dimensional scalar field theory.