Discrete Orthogonality Relations Research Articles

Discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts

The discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts are investigated. We show that the discrete orthogonality relations hold for the case-(1) multi-indexed orthogonal polynomials of continuous Hahn, Wilson, and Askey–Wilson types, and we conjecture their normalization constants.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓD≥1, the three term recurrence relations are broken. The extra ℓD “lower degree polynomials,” which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree ℓD+N. The discrete orthogonality relations are shown to hold for another group of “new” orthogonal polynomials called Krein–Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials.

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .

Discretization of Generalized Chebyshev Polynomials of (Anti)symmetric Multivariate Sine Functions

The multivariate antisymmetric and symmetric trigonometric functions allow to generalize the four kinds of classical Chebyshev polynomials to multivariate settings. The four classes of the bivariate polynomials, related to the symmetrized sine functions, are studied in detail. For each of these polynomials, the weighted continuous and discrete orthogonality relations are shown. The related cubature formulas for numerical integration together with further model examples and properties of selected special cases are discussed.

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified.

On E–discretization of tori of compact simple Lie groups. II

Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine, and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgroup of the Weyl group. Fundamental domains of even affine and dual even affine Weyl groups, governing the argument and label symmetries of the even orbit functions, are determined. The discrete orthogonality relations are formulated on finite sets of points from the refinements of the dual weight lattices. Explicit counting formulas for the number of points of the discrete transforms are deduced. Real-valued Hartley orbit functions are introduced, and all ten types of the corresponding discrete Hartley transforms are detailed.

Orthogonality of Macdonald polynomials with unitary parameters

For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation relation.

Orthogonal Basic Hypergeometric Laurent Polynomials

The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$, which are given as a sum of two terminating $_4\phi_3$'s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single $_4\phi_3$'s which are Laurent polynomials in $z$ are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.

Finite-Dimensional Orthogonality Structures for Hall-Littlewood Polynomials

We present a finite-dimensional system of discrete orthogonality relations for the Hall-Littlewood polynomials. A compact determinantal formula for the weights of the discrete orthogonality measure is formulated in terms of a Gaudin-type conjecture for the normalization constants of a dual system of orthogonality relations. The correctness of our normalization conjecture has been checked in some special cases: for Hall-Littlewood polynomials up to four variables (i), for the reduction to Schur polynomials (ii), and in a continuum limit in which the Hall-Littlewood polynomials degenerate into the Bethe Ansatz eigenfunctions of the Schrodinger operator for identical Bose particles on the circle with pairwise delta-potential interactions (iii).

A generalization of the prolate spheroidal wave functions

Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of L 2 ( − ∞ , ∞ ) L^2(-\infty , \infty ) and L 2 ( 0 , ∞ ) , L^2(0,\infty ), and the Jacobi polynomials which are an orthogonal basis of a weighted L 2 ( − 1 , 1 ) . L^2(-1,1). The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of L 2 ( − 1 , 1 ) . L^2(-1,1). The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both L 2 ( − 1 , 1 ) L^2(-1,1) and a subspace of L 2 ( − ∞ , ∞ ) , L^2(-\infty , \infty ), known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property. The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group $G$. Such functions can always be restricted without loss of information to a fundamental region $\check F$ of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over $\check F$ (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in $\check F$ (discrete orthogonality), arising as the set of points in $\check F$ representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus $\mathbb T$. The characters of the centre $Z$ of the Lie group allow one to split functions $f$ on $\check F$ into a sum $f=f_1+...+f_c$, where $c$ is the order of $Z$, and where the component functions $f_k$ decompose into the series of $C$-, or $S$-, or $E$-functions from one congruence class only.

LU factorizations, q = 0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas.

Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials

By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials P n ( 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 P n ( x ; q α , q α , − q α ; 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 P n ( x ; a , a , − a ; q ) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.

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.

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.

Uniform Asymptotic Expansions for Charlier Polynomials

The asymptotic behaviour of the Charlier polynomials C(a)n(x) as n→∞ is examined. These polynomials satisfy a discrete orthogonality relation and, unlike classical orthogonal polynomials, do not satisfy a second-order linear differential equation with respect to the independent variable x. As such, previous results on their asymptotic behaviour have been restricted to integral methods and consequently have been quite limited in their scope. In this paper a new approach is used, where the polynomials C(a)n(x) are not regarded as a functions of x with a as a parameter, but rather with the roles reversed via a second-order linear differential equation in which a is the (real or complex-valued) independent variable and x is a parameter. This equation has two turning points in the a plane which depend on x, and are either positive or complex conjugates, according to the values of x. Moreover, the turning points can coalesce with one another, or one with a singularity of the equation, for certain critical values of x. By using two general asymptotic theories of differential equations, one for intervals free of turning points and the other for intervals containing a double pole and a coalescing turning point, expansions are derived for C(a)n(x) involving either elementary functions or Bessel functions. Taken together, the results are uniformly valid for −∞<x<∞. In addition, in each case the expansions are uniformly valid for a lying in certain unbounded intervals, each of which contain [−{1−δ} n, {1−δ} n], where δ∈(0, 1) is an arbitrary constant.

Some Explicit Padé Approximants for the Function ${{\Phi '} / \Phi }$ and a Related Quadrature Formula Involving Bessel Functions

In this paper we determined in closed form the $[n\mid n]$ Padé approximant for the logarithmic derivative of the confluent hypergeometric function of the first kind, and also an explicit formula for the error. We next show how the recurrence defining the numerators and denominators of the approximants can be used to deduce a certain discrete orthogonality relationship. A consequence of this is a discrete orthogonality relation for the Bessel function of the first kind and an exact quadrature formula involving this function.

Generalized n-gons and Chebychev polynomials

Two sequences of orthogonal polynomials are given whose weight functions consist of an absolutely continuous part and two point masses. Combinatorial proofs of the orthogonality relations are given. The polynomials include natural q -analogs of the Chebychev polynomials. The technique uses association schemes of generalized n -gons to find approximating discrete orthogonality relations. The Feit-Higman Theorem is a corollary of these orthogonality relations for the polynomials.