Continuous Dual Hahn Polynomials Research Articles

Characterization of orthogonal polynomials on lattices

We consider two sequences of orthogonal polynomials and such that with , and are sequences of complex numbers, , is the identity operator, x defines a lattice, and . We show that under some natural conditions, both involved orthogonal polynomials sequences and are semiclassical whenever k = m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.

Moments of orthogonal polynomials and exponential generating functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( $$q=1$$ ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials.

On the continuous dual Hahn process

We extend the continuous dual Hahn process (Tt) of Corwin and Knizel from a finite time interval to the entire real line by taking a limit of a closely related Markov process (Tt). We also characterize processes (Tt) by conditional means and variances under bidirectional conditioning, and we prove that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes.

A class of Sobolev orthogonal polynomials on the unit circle and associated continuous dual Hahn polynomials: Bounds, asymptotics and zeros

This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with respect to these parameters. Criteria for knowing when the zeros of the (Sobolev) orthogonal polynomials and also the zeros of their derivatives stay within the unit disk have also been addressed. By numerical experiments some further information on the parameters is also found so that the zeros remain within the unit disk.

Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

AbstractWe derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

ABSTRACTZeilberger's celebrated algorithm finds pure recurrence relations (w.r.t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and special functions, contiguous relations w.r.t. several variables exist in abundance. We modify Zeilberger's algorithm to generate unknown contiguous relations that are necessary to obtain inner bounds for the extreme zeros of orthogonal polynomial sequences with hypergeometric representations. Using this method, we improve previously obtained upper bounds for the smallest and lower bounds for the largest zeros of the Hahn polynomials and we identify inner bounds for the extreme zeros of the Continuous Hahn and Continuous Dual Hahn polynomials. Numerical examples are provided to illustrate the quality of the new bounds. Without the use of computer algebra such results are not accessible. We expect our algorithm to be useful to compute useful and new contiguous relations for other hypergeometric functions.

Quasi-orthogonality of some hypergeometric polynomials

ABSTRACTThe zeros of quasi-orthogonal polynomials play a key role in applications in areas such as interpolation theory, Gauss-type quadrature formulas, rational approximation and electrostatics. We extend previous results on the quasi-orthogonality of Jacobi polynomials and discuss the quasi-orthogonality of Meixner–Pollaczek, Hahn, Dual-Hahn and Continuous Dual-Hahn polynomials using a characterization of quasi-orthogonality due to Shohat. Of particular interest are the Meixner–Pollaczek polynomials whose linear combinations only exhibit quasi-orthogonality of even order. In some cases, we also investigate the location of the zeros of these polynomials for quasi-orthogonality of order 1 and 2 with respect to the end points of the interval of orthogonality, as well as with respect to the zeros of different polynomials in the same orthogonal sequence.

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

A three-parameter family of weighted Hankel matrices is introduced with the entries, supposing , , are positive and , , . The famous Hilbert matrix is included as a particular case. The direct sum is shown to commute with a discrete analogue of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, , commuting with . The orthogonal polynomials associated with turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping diagonalizing can be constructed explicitly. At the same time, diagonalizes and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval where is known explicitly. If the assumption is relaxed while the remaining inequalities on , , are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold . Again, all eigenvalues and eigenvectors are described explicitly.

The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

A slight modification of the Kontorovich–Lebedev transform is an auto‐morphism on the vector space of polynomials. The action of this ‐transform over certain polynomial sequences will be under discussion, and a special attention will be given to the d‐orthogonal ones. For instance, the Continuous Dual Hahn polynomials appear as the ‐transform of a 2‐orthogonal sequence of Laguerre type. Finally, all the orthogonal polynomial sequences whose ‐transform is a d‐orthogonal sequence will be characterized: they are essencially semiclassical polynomials fulfilling particular conditions and d is even. The Hermite and Laguerre polynomials are the classical solutions to this problem.

Deformed su(1,1) Algebra as a Model for Quantum Oscillators

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra $\mathfrak{su}(1,1)_\gamma$. Just as the positive discrete series representations of $\mathfrak{su}(1,1)$ can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of $\mathfrak{su}(1,1)_\gamma$ can be utilized to construct models of a quantum oscillator. In this case, the wave functions are expressed in terms of continuous dual Hahn polynomials. We study some properties of these wave functions, and illustrate some features in plots. We also discuss some interesting limits and special cases of the obtained oscillator models.

PROBABILITY MEASURES ON ℂ ARISING FROM THE JACOBI–SZEGÖ PARAMETERS FOR CONTINUOUS DUAL HAHN POLYNOMIALS

In this paper, we shall construct probability measures on ℂ, which can be expressed by the Mellin convolution product of the modified Bessel functions. Our measures on ℂ are related to the Jacobi–Szegö parameters for the continuous dual Hahn polynomials Sn(x2, a, b, c) under the special choice of parameters a, b, c. This paper contains new materials which go further than our previous results in Ref.7. The most interesting thing is that the Mellin convolution of two modified Bessel functions can again be expressed in terms of the modified Bessel functions, by choosing parameters a, b, c appropriately. The origin of our research in this direction goes back to the Bargmann–Fock representation of the classical non-Gaussian random variables.5–8 Our results would have a potential to be related to the higher powers of creation and annihilation operators acting on a new class of interacting Fock spaces.

Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the well-known exactly solvable difference equations of the Meixner-Pollaczek, continuous Hahn, continuous dual Hahn, Wilson and Askey-Wilson polynomials. Up to an overall factor of the so-called pseudo ground state wavefunction, the eigenfunctions within the exactly solvable subspace are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots.

THE CONSTRUCTION OF SUBORDINATED PROBABILITY MEASURES ON ℂ ASSOCIATED WITH THE JACOBI–SZEGÖ PARAMETERS

In this paper, it will be shown that a probability measure on ℂ associated with the Jacobi–Szegö parameters of the orthogonal polynomials can be obtained by making use of the classical Mellin transform and its convolution property. We shall construct several measures on ℂ represented by the modified Bessel functions. The material in this paper gives nontrivial examples originated from the continuous dual Hahn polynomials (one of hypergeometric orthogonal polynomials), which are beyond the Meixner–Pollaczek polynomials appeared in our previous papers.4, 5

SUMS OF SQUARES FROM ELLIPTIC PFAFFIANS

We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m2and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's formulas to Schur Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m2squares.

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), Lopez and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.

Branching laws for minimal holomorphic representations

In this paper we study the branching law for the restriction from SU ( n , m ) to SO ( n , m ) of the minimal representation in the analytic continuation of the scalar holomorphic discrete series. We identify the group decomposition with the spectral decomposition of the action of the Casimir operator on the subspace of S ( O ( n ) × O ( m ) ) -invariants. The Plancherel measure of the decomposition defines an L 2 -space of functions, for which certain continuous dual Hahn polynomials furnish an orthonormal basis. It turns out that the measure has point masses precisely when n − m > 2 . Under these conditions we construct an irreducible representation of SO ( n , m ) , identify it with a parabolically induced representation, and construct a unitary embedding into the representation space for the minimal representation of SU ( n , m ) .

Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases

We write spectral decomposition of the hypergeometric differential operator on the contour $Re z=1/2$ (multiplicity of spectrum is 2). As a result, we obtain an integral transform that differs from the Jacobi (or Olevsky) transform. We also try to do a step towards vector-valued special functions and construct a 3F2 orthogonal basis in the space of functions having values in 2-dimensional space. This basis is lying in an analytic continuation of continuous dual Hahn polynomials with respect to number $n$ of a polynomial. In Addendum, we discuss a vector-value analog of the Meixner--Pollachek orthogonal system and also perturbations of Laguerre, Meixner, and Jacobi polynomials.