Continuous Dual Hahn Polynomials Research Articles

Square ice, alternating sign matrices, and classical orthogonal polynomials

The six-vertex model with domain wall boundary conditions, or squareice, is considered for particular values of its parameters, corresponding to1-,2-,and 3-enumerations of alternating sign matrices (ASMs). Using Hankel determinant representationsfor the partition function and the boundary correlator of homogeneous square ice, it is shownhow the ordinary and refined enumerations can be derived in a very simple and straightforwardway. The derivation is based on the standard relationship between Hankel determinantsand orthogonal polynomials. For the particular sets of parameters corresponding to1-,2-,and 3-enumerations of ASMs, the Hankel determinant can be naturally related to ContinuousHahn, Meixner–Pollaczek, and Continuous Dual Hahn polynomials, respectively. Thisobservation allows for a unified and simplified treatment of ASM enumerations.In particular, along the lines of the proposed approach, we provide a completesolution to the long-standing problem of the refined 3-enumeration of ASMs.

A relativistic model of the isotropic three-dimensional singular oscillator

Exactly solvable model of the quantum isotropic three-dimensional singular oscillator in the relativistic configurational r → -space is proposed. We have found the radial wavefunctions, which are expressed through the continuous dual Hahn polynomials and energy spectrum for the model under consideration. It is shown that they have the correct non-relativistic limits.

L2series solutions of the Dirac equation for power-law potentials at rest mass energy

We obtain solutions of the three dimensional Dirac equation for radial power-law potentials at rest mass energy as an infinite series of square integrable functions. These are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the Dirac operator is tridiagonal. The "wave equation" results in a three-term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials. These are modified versions of the Meixner-Pollaczek polynomials and of the continuous dual Hahn polynomials. The choice depends on the values of the angular momentum and the power of the potential.

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.

Tensor products of Minimal Holomorphic Representations

LetD=G/KD=G/Kbe an irreducible bounded symmetric domain with genusppandHν(D)H^{\nu }(D)the weighted Bergman spaces of holomorphic functions forν>p−1\nu >p-1. The spacesHν(D)H^\nu (D)form unitary (projective) representations of the groupGGand have analytic continuation inν\nu; they give also unitary representations whenν\nuin the Wallach set, which consists of a continuous part and a discrete part ofrrpoints. The first non-trivial discrete pointν=a2\nu =\frac a2gives the minimal highest weight representation ofGG. We give the irreducible decomposition of tensor productHa2⊗Ha2¯H^{\frac a2}\otimes \overline {H^{\frac a2}}. As a consequence we discover some new spherical unitary representations ofGGand find the expansion of the corresponding spherical functions in terms of theKK-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.

Integral Representations of the Wilson Polynomials and the Continuous Dual Hahn Polynomials

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.

Multivariable Wilson polynomials

A multivariable biorthogonal generalization of the Wilson polynomials is presented. These are four distinct families, which in a special case occur in two complex conjugate pairs, that satisfy four biorthogonality relations among them. An interesting limit case is the multivariable continuous dual Hahn polynomials.

Quadratic Birth and Death Processes and Associated Continuous Dual Hahn Polynomials

Birth and death process polynomials with symmetric quadratic rates are studied. They provide two generalizations of the continuous dual Hahn polynomials. Generating functions and explicit representations are derived. The asymptotic behavior and weight functions of the polynomials under consideration are also determined.