Second-order Q-difference Equations Research Articles

The uniqueness of meromorphic function shared values with meromorphic solutions of a class of q-difference equations

<abstract><p>We first investigate the meromorphic solutions of a class of homogeneous second-order q-difference equations and the uniqueness problem for a meromorphic function with three shared values; then we discuss the uniqueness problem for the meromorphic solutions of a class of nonhomogeneous q-difference equations and a meromorphic function with four shared values.</p></abstract>

Existence of positive strongly decaying solutions of second-order nonlinear q-difference equations

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying and as are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Two Finite Sequences of Symmetric q-Orthogonal Polynomials Generated by Two q-Sturm–Liouville Problems

By using a symmetric generalization of Sturm–Liouville problems in q-difference spaces, we introduce two finite sequences of symmetric q-orthogonal polynomials and obtain their basic properties such as a second-order q-difference equations, the explicit form of the polynomials in terms of basic hypergeometric series, three-term recurrence relations and norm-square values based on a Ramanujan identity. We also show that one of the introduced sequences is connected with the little q-Jacobi polynomials.

Existence results for nonlinear second-order q-difference equations with q-integral boundary conditions

Existence results for nonlinear second-order q-difference equations with q-integral boundary conditions

Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives

In this paper, we establish the existence of solutions for a boundary value problem with the nonlinear second-order q-difference equation { D q 2 u ( t ) = f ( t , u ( t ) , D q u ( t ) ) , t ∈ I , D q u ( 0 ) = 0 , D q u ( 1 ) = α u ( 1 ) . The existence and uniqueness of solutions for the problem are proved by means of the Leray-Schauder nonlinear alternative and some standard fixed point theorems. Finally, we give two examples to demonstrate the use of the main results. The nonlinear team f contains D q u(t) in the equation.

On q-Difference Riccati Equations and Second-Order Linear q-Difference Equations

We consider q-difference Riccati equations and second-order linear q-difference equations in the complex plane. We present some basic properties, such as the transformations between these two equations, the representations and the value distribution of meromorphic solutions of q-difference Riccati equations, and the q-Casorati determinant of meromorphic solutions of second-order linear q-difference equations. In particular, we find that the meromorphic solutions of these two equations are concerned with the q-Gamma function when q∈ℂ such that 0<|q|<1. Some examples are also listed to illustrate our results.

A study of second-order q-difference equations with boundary conditions

This paper studies a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the q-difference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of q-difference equations. The results corresponding to a second-order q-difference equation with anti-periodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical second-order boundary value problems with non-separated boundary conditions in the limit q → 1, which provides a useful check.2010 Mathematics Subject Classification. 39A05, 39A13.

Structure relations for q-polynomials and some applications

We derive structure relations for polynomials orthogonal on a half-line or on the real line. Among other things, we derive their degree raising and lowering first-order q-difference operators. We study the properties of the basis of solutions of the corresponding second-order q-difference equation. This generalizes the results of Ismail and Simeonov [M.E.H. Ismail and P. Simeonov, q-difference operators for orthogonal polynomials, J. Comput. Appl. Math. 233 (3) (2009), 749–761]. We apply these structure relations and similar known ones in differential equations to derive the nonlinear difference equations satisfied by the sequence {β n }, where β n are the coefficients of the three-term recurrence relation satisfied by orthogonal polynomials. The polynomials under consideration are orthogonal with respect to q-analogues of exponential weights (Freud weights).

Q-regular variation and q-difference equations

We introduce the concept of q-regularly varying functions and establish basic properties of such functions. Among other things it is shown that considering regular variation in q-calculus is somehow natural and leads to interesting observations and simplifications compared with classical continuous and discrete theories. The obtained theory is applied to an investigation of asymptotic behavior of solutions to linear second-order q-difference equations.

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 q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

The spectral properties of operators formed from generators of the q-Onsager non-Abelian infinite-dimensional algebra are investigated. Using a suitable functional representation, all eigenfunctions are shown to obey a second-order q-difference equation (or its degenerate discrete version). In the algebraic sector associated with polynomial eigenfunctions (or their discrete analogues), Bethe equations naturally appear. Beyond this sector, where the Bethe ansatz approach is not applicable in related massive quantum integrable models, the eigenfunctions are also described. The spin-half XXZ open spin chain with general integrable boundary conditions is reconsidered in light of this approach: all the eigenstates are constructed. In the algebraic sector which corresponds to special relations among the parameters, known results are recovered.

Kneser's theorem inq-calculus

While difference equations deal with discrete calculus and differential equations with continuous calculus, so-called q-difference equations are considered when studying q-calculus. In this paper, we obtain certain oscillation criteria for second-order q-difference equations, among them a q-calculus version of the famous Kneser theorem.

Hypergeometric type q-difference equations: Rodrigues type representation for the second kind solution

A Rodrigues type representation for the second kind solution of a second-order q-difference equation of hypergeometric type is given. This representation contains some q-extensions of integrals related with relevant special functions. For these integrals, a general recurrence relation, which only involves the coefficients of the q-difference equation, is also presented.

Difference equations and quantized discriminants for q-orthogonal polynomials

We show that orthogonal polynomials on generalized q-linear grid have raising and lowering operators and satisfy a second-order q-difference equation. It is shown that for a general class of weight functions on general q-linear grids, the functions of the second kind and the orthogonal polynomials are linear independent solutions of the same second-order q-difference equation. We introduce q-analogues of the discriminant and evaluate the quantized discriminant for general q-orthogonal polynomials in terms of the recursion coefficients. The q-discriminants of the discrete q-Hermite and little q-Jacobi polynomials, as well as a generalized discriminant of the continuous q-Jacobi polynomials, are given explicitly. A q-analogue of the Freud weights is introduced and we derive the second-order q-difference equation they satisfy.

The factorization method for the general second-orderq-difference equation and the Laguerre Hahn polynomials on the generalq-lattice

For the general linear second-order q-difference equation, we show the interconnection between the factorization method and the Laguerre–Hahn polynomials on the general q-lattice. Applications are then given in the cases of the hypergeometric and Askey–Wilson second-order q-difference equations.

Difference equations for the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre–Hahn class

We use some relations between the rth associated orthogonal polynomials of the Dq-Laguerre–Hahn class to derive the fourth-order q-difference equation satisfied by the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre–Hahn class.When r=1 and for q-semi-classical situations, this q-difference equation factorizes as product of two second-order q-difference equations. Finally, we study some classical situations, and give some examples relative to the co-recursive associated discrete q-Hermite II orthogonal polynomials.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

The second-order q-difference equation of a 2 ϑ 1 and the contiguous relations of 2 ϑ 1's are used to derive continued fraction expansions for quotients of 2 ϑ 1's. We identify the cases when these continued fractions are positive definite J-fractions, i.e., the cases when the denominators of the approximants of the J-fractions are orthogonal polynomials. The measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail.