On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials

We derive discrete orthogonality relations for polynomials, dual to little and big q-Jacobi polynomials. This derivation essentially requires use of bases, consisting of eigenvectors of certain self-adjoint operators, which are representable by a Jacobi matrix. Recurrence relations for these polynomials are also given.     

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})\) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.     

On Jacobi and continuous Hahn polynomials

Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform, and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the Parseval formula. In a special case this relation dates back to work by Bateman in 1933 and we follow a part of the historical development for these polynomials. Some applications of this relation are given.

     

An algebraic interpretation of the multivariate <Emphasis Type="Italic">q</Emphasis>-Krawtchouk polynomials

The multivariate quantum q-Krawtchouk polynomials are shown to arise as matrix elements of “q-rotations” acting on the state vectors of many q-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations, and recurrence relations. The extension to an arbitrary number of variables is presented.     

Quadratures with multiple nodes,power orthogonality,and moment-preserving spline approximation

Quadrature formulas with multiple nodes, power orthogonality, and some applications of such quadratures to moment-preserving approximation by defective splines are considered. An account on power orthogonality (s- and σ-orthogonal polynomials) and generalized Gaussian quadratures with multiple nodes, including stable algorithms for numerical construction of the corresponding polynomials and Cotes numbers, are given. In particular, the important case of Chebyshev weight is analyzed. Finally, some applications in moment-preserving approximation of functions by defective splines are discussed.     

Extraction of orthogonal polynomials from generating function for reciprocal of odd numbers

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both even and odd order convergents of a regular C-fraction of a power series with coefficients as reciprocal of odd numbers are described. The two sequences of convergents are nothing but diagonal and upper diagonal Pade approximants for the power series. The two orthogonal polynomials extracted from denominators are shown to be classical orthogonal polynomials and two orthogonal polynomials extracted from numerators are shown to be non-classical orthogonal polynomials..     

A q-summation formula,the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities.     

A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials

The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials, the operator T can also be realized as a five-diagonal operator, leading to orthogonality relations for 2×2-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.     

A special class of polynomials orthogonal on the unit circle including the associated polynomials

Let (P _ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ω_υ) the monic associated polynomials of (P _v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (AP_υλ+BΩ_υλ)/A^λ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP _ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP _ν and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.     

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.     

The Clebsch-Gordan coefficients for the quantum group SμU(2) and q-Hahn polynomials

The tensor product of two unitary irreducible representations of the quantum group S_μU(2) is decomposed in a direct sum of unitary irreducible representations with explicit realizations. The Clebsch-Gordan coefficients yield the orthogonality relations for q-Hahn and dual q-Hahn polynomials.     

Pseudo-Orthogonal Polynomials

We consider the classical problem of transforming an orthogonality weight of polynomials by means of the space R _n . We describe systems of polynomials called pseudo-orthogonal on a finite set of n points. Like orthogonal polynomials, the polynomials of these systems are connected by three-term relations with tridiagonal matrix which is nondecomposable but does not enjoy the Jacobi property. Nevertheless these polynomials possess real roots of multiplicity one; moreover, almost all roots of two neighboring polynomials separate one another. The pseudo-orthogonality weights are partly negative. Another result is the analysis of relations between matrices of two different orthogonal systems which enables us to give explicit conditions for existence of pseudo-orthogonal polynomials.     

On type basic hypergeometric orthogonal polynomials

The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable -Racah polynomials and multivariable big and little -Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known -analogues of the Selberg integral are obtained.

     

Orthogonal polynomials and laurent polynomials related to the Hahn-Extonq-Bessel function

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.     

On differential properties for bivariate orthogonal polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.      

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra su(l, 1). The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of su(l,l) are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by an integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.     

A certain class of q-Bessel polynomials

A q-extension of the familiar Bessel polynomials is considered here from the point of view of their associated q-differential equation. The orthogonality of these q-Bessel polynomials is discussed. Several remarks and observations, relevant to the present investigation, are also made.     

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.