Q-orthogonal Polynomials Research Articles

Description of the symmetric $$H_q$$-Laguerre–Hahn orthogonal q-polynomials of class one

Description of the symmetric $$H_q$$-Laguerre–Hahn orthogonal q-polynomials of class one

Some characterizations of the symmetric q-Dunkl-semiclassical orthogonal q-polynomials

In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion of a class and a criterion for determining the class will be given. As an application, several examples of q-Dunkl-semiclassical forms of the class s ≦ 2 are highlighted.

Asymptotics of discrete q-Freud orthogonal polynomials from the q-Riemann Hilbert problem

We investigate a Riemann–Hilbert problem (RHP), whose solution corresponds to a group of q-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the q-discrete lattice and determine the asymptotics of a related q-Painlevé equation.

Asymptotic Behaviours of q-orthogonal Polynomials from a q-Riemann Hilbert Problem

We describe a Riemann-Hilbert problem for a family of $q$-orthogonal polynomials, $\{ P_n(x) \}_{n=0}^\infty$, and use it to deduce their asymptotic behaviours in the limit as the degree, $n$, approaches infinity. We find that the $q$-orthogonal polynomials studied in this paper share certain universal behaviours in the limit $n\to\infty$. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, $x \sim q^{n/2}$, and norm, $\|P_n\|_2$, are independent of the weight function as $n\to\infty$.

Rogers–Ramanujan type identities and Chebyshev polynomials of the third kind

It is known that q-orthogonal polynomials play an important role in the field of q-series and special functions. While studying Dyson’s “favorite” identity of Rogers–Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of q. By introducing Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers–Ramanujan type identities and also results related to mock theta functions and Hecke-type series. In this paper, by constructing a new Bailey pair involving Chebyshev polynomials of the third kind, we further extend Andrews’ way of studying Rogers-Ramanujan type identities. By inserting this Bailey pair into various weak forms of Bailey’s lemma, we obtain a companion identity for Dyson’s favorite identity and a number of Rogers–Ramanujan type identities. As a consequences, we also obtain results related to Appell–Lerch series and the generalized Hecke-type series. Furthermore, our key Bailey pair also fits in the bilateral versions of Bailey’s lemma due to Andrews and Warnaar, which leads to more identities for the generalized Hecke-type series and false theta functions.

A Littlewood–Richardson rule for Koornwinder polynomials

Koornwinder polynomials are q-orthogonal polynomials equipped with extra five parameters and the $$B C_n$$ -type Weyl group symmetry, which were introduced by Koornwinder (Contemp Math 138:189–204, 1992) as multivariate analogue of Askey–Wilson polynomials. They are now understood as the Macdonald polynomials associated with the affine root system of type $$(C^\vee _n,C_n)$$ via the Macdonald–Cherednik theory of double affine Hecke algebras. In this paper, we give explicit formulas of Littlewood–Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural $$(C^\vee _n,C_n)$$ -analogue of Yip’s alcove-walk formulas (Math Z 272:1259–1290, 2012) which were given in the case of reduced affine root systems.

Characterizations of the Symmetric $$T_{(\theta , q)}$$-Classical Orthogonal q-Polynomials

Characterizations of the Symmetric $$T_{(\theta , q)}$$-Classical Orthogonal q-Polynomials

Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials

In this paper, we show that, up to a dilatation, the $$q^2$$ -analogue of generalized Hermite and $$q^2$$ -analogue of generalized Gegenbauer polynomials are the only q-Dunkl-classical symmetric orthogonal polynomials.

On Fractional q-Extensions of Some q-Orthogonal Polynomials

In this paper, we introduce a fractional q-extension of the q-differential operator Dq−1 and prove some of its main properties. Next, fractional q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly-defined functions are given. Finally, a fractional q-difference equation of Gaussian type is introduced and solved by means of the power series method.

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.

A fractional q-derivative operator and fractional extensions of some q-orthogonal polynomials

A fractional q-derivative operator is introduced and some of its properties have been proved. Next, a fractional q-differential equation of Gauss type is introduced and solved by means of a power series method. Finally, q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly defined functions are given.

A note on q-difference equations for Ramanujan’s integrals

This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to obtain the expression of certain q-orthogonal polynomials as solutions of q-difference equations. In addition, we show how to generalize Ramanujan’s integrals by the technique of q-difference equation. More over, we find two generalized q-Chu–Vandermonde formulas from the perspective of the method of q-difference equations.

Mixed recurrence equations and interlacing properties for zeros of sequences of classical q-orthogonal polynomials

Using the q-version of Zeilberger's algorithm, we provide a procedure to find mixed recurrence equations satisfied by classical q-orthogonal polynomials with shifted parameters. These equations are used to investigate interlacing properties of zeros of sequences of q-orthogonal polynomials. In the cases where zeros do not interlace, we give some numerical examples to illustrate this.

Divided-difference equation and three-term recurrence relations of some systems of bivariate q-orthogonal polynomials

Partial divided-difference equations and three-term recurrence relations satisfied by the bivariate Askey–Wilson and the bivariate q-Racah polynomials are computed in this work. By using limiting processes, partial divided (or q)-difference equations and three-term recurrence relations are also provided for each of the following families of orthogonal polynomials: the bivariate continuous dual q-Hahn, the bivariate Al-Salam-Chihara, the bivariate continuous q-Hahn, the bivariate q-Hahn, the bivariate dual q-Hahn, the bivariate q-Krawtchouk, the bivariate q-Meixner, and the bivariate q-Charlier polynomials.

Q-Analogue modified Laguerre and generalized modified Laguerre Polynomials of One Variable

The q-Laguerre polynomials are important q-orthogonal polynomials whose applications and generalizations arise in many applications such as quantumgroup (oscillator algebra, etc.), q-harmonic oscillator and coding theory. In this paper, we introduce the q-analogue modified Laguerre polynomials and generalized modified Laguerre polynomials of one variable. Some recurrence relations and q-Laplace transforms for these polynomials are derived.

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.

On some 2𝐷 orthogonal 𝑞-polynomials

We introduce two q q -analogues of the 2 D 2D -Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a q q - 2 D 2D analogue of the disk polynomials (Zernike polynomials) and derive similar formulas for them as well, including evaluating certain connection coefficients. Some of the generating functions may be related to Rogers–Ramanujan type identities.

A finite class of q-orthogonal polynomials corresponding to inverse gamma distribution

In this paper, a class of finite q-orthogonal polynomials is studied whose weight function corresponds to the inverse gamma distribution as \(q \rightarrow 1\). Via Sturm–Liouville theory in q-difference spaces, the orthogonality of this class is proved and its norm square value is computed. Also, its general properties such as q-weight function, q-difference equation and the basic hypergeometric representation are recovered in the continuous case.

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.

Solvable discrete quantum mechanics: q-orthogonal polynomials with q=1 and quantum dilogarithm

Several kinds of q-orthogonal polynomials with q=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions, which are a natural extension of the Euler gamma functions and the q-gamma functions (q-shifted factorials). The dimensions of the orthogonal spaces are finite. These q-orthogonal polynomials are expressed in terms of the Askey-Wilson polynomials and their certain limit forms.