Terms Of Classical Orthogonal Polynomials Research Articles

An urn model for the Jacobi-Piñeiro polynomials

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.

On the Bombieri numbers for the class S

We derive a second variation formula of the Koebe function developed by Bombieri by using a linear version of the Loewner differential equation. We then express the formula in terms of classical orthogonal polynomials to consider the Bombieri numbers obtained by Greiner and Roth (Proc Am Math Soc 129(12):3657–3664, 2001) and by Prokhorov and Vasil’ev (Georgian Math J 12(4):743–761, 2005). We show a question raised by Bombieri on these numbers has a negative answer in the low order cases.

A generalized quantum nonlinear oscillator

We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the new exactly solvable potentials possess shape invariance symmetry. The solutions are obtained in terms of classical orthogonal polynomials.

Sharp estimates for the convergence rate of Fourier series in terms of orthogonal polynomials in L 2((a, b), p(x))

Sharp estimates are given for the convergence rate of Fourier series in terms of classical orthogonal polynomials in some classes of functions characterized by a generalized modulus of continuity in the space L 2((a, b), p(x)). Expansions in terms of Laguerre, Hermite, and Jacobi polynomials are considered.

Eigenvalue separation in some random matrix models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogs, an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

Classical orthogonal polynomials and Leverrier-Faddeev algorithm for the matrix pencilssE−A

In this contribution we present an extension of the Leverrier-Faddeev algorithm for the simultaneous computation of the determinant and the adjoint matrixB(s)of a pencilsE−AwhereEis a singular matrix butdet⁡(sE−A)≢0. Using a previous result by the authors we expressB(s)anddet⁡(sE−A)in terms of classical orthogonal polynomials.

Linear stability of source-type similarity solutions of the thin film equation

We study the stability of compactly-supported source-type self-similar solutions of the generalized thin film equation h t = −( h n h xxx ) x . Using linear stability analysis, applied to the problem in similarity variables, we show that the source-type solutions are stable. These results are related to the underlying symmetries of the PDE. For the special case of n = 1, analytical results are obtained for the spectrum, and the eigenfunctions are given in terms of classical orthogonal polynomials.

Quantum superintegrability and exact solvability in n dimensions

A family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in n-dimensional Euclidean space. Two different sets of n commuting second-order operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.

Decomposition of Polynomials with Respect to the Cyclic Group of Orderm

Given a polynomial solution of a differential equation, its m -ary decomposition, i.e. its decomposition as a sum of m polynomials P[ j ](x) =∑kαj,kxλj, kcontaining only exponentsλj, k with λj,k+1−λj,k=m, is considered. A general algorithm is proposed in order to build holonomic equations for the m -ary parts P[ j ](x) starting from the initial one, which, in addition, provides a factorized form of them. Moreover, these differential equations are used to compute expansions of the m -ary parts of a given polynomial in terms of classical orthogonal polynomials. As illustration, binary and ternary decomposition of these classical families are worked out in detail.

A Multi-dimensional Linear Growth Birth and Death Process

In [5] Karlin and McGregor analyzed a class of linear growth birth and death processes admitting a representation formula of the transition probability function in terms of classical orthogonal polynomials. Birth and death processes belong to the category of reversible stochastic processes (see [12] and [13]). In this case by invoking the spectral resolution of the identity for Hermitian operators one achieves the representation formula of Karlin and McGregor described in [3] and [4]. This theory is mostly restricted to the case of one-dimensional birth and death processes and diffusion processes. Some special higher dimensional birth and death processes motivated by certain applications in studies of population growth are reversible. For these processes there are representation formulae for the transition probability function that can be explicitly determined. In [9] and [10] Karlin and McGregor develop a discretized version of the classical technique of solving Laplace's equation in terms of spherical harmonics. This paper applies their method to determine the representation formula for the transition probability function of two two-dimensional (Sections 2 and 3) and the corresponding higher dimensional (Sections 4 and 5) linear growth birth and death processes. In each case the representation formula is expressed explicitly in terms of semi-direct products of classical orthogonal polynomials somewhat reminiscent of such products of spherical harmonics. In the Appendix the properties of the orthogonal polynomials are summarized and the method of derivation of the explicit form of the representation formula is outlined.