Characterizations of some polynomial variance functions byd-pseudo-orthogonality

From a notion ofd-pseudo-orthogonality for a sequence of polynomials (d ∈ {2,3, …}), this paper introduces three different characterizations of natural exponential families (NEF's) with polynomial variance functions of exact degree 2d-1. These results provide extended versions of the Meixner (1934), Shanbhag (1972), 1979) and Feinsilver (1986) characterization results of quadratic NEF's based on classical orthogonal polynomials. Some news sets of polynomials with (2d-1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions.     

Wahrscheinlichkeitsfunktionen diskreter Verteilungen als Lösungen der Pearsonschen Differenzengleichung für die diskreten klassischen Orthogonalpolynome

Using the Pearson difference equation for the discrete classical orthogonal polynomials the difference equations and the Rodrigues formulas are obtained. The resulting weight functions prove to be the probability functions of the most important discrete probability distributions: Pólya distribution from the Hahn and Krawtchouk polynomials, negative binomial distribution from the Meixner polynomials, Poisson distribution from the Charlier polynomials.     

Fourth-order difference equation for the associated Meixner and Charlier polynomials

An explicit representation of the associated Meixner polynomials (with a real association parameter γ?0) is given in terms of hypergeometric functions. This representation allows to derive the fourth-order difference equation verified by these polynomials. Appropriate limits give the fourth-order difference equation for the associated Charlier polynomials and the fourth-order differential equations for the associated Laguerre and Hermite polynomials.     

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials.     

Free martingale polynomials

In this paper we investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.     

Moment systems and orthogonal polynomials in several variables

The first part of this paper deals with general moment (“Appell”) systems on $\text{R}$^N generated by a Hamiltonian function H(x, D) and also with representations of GL(N) on the associated spaces of polynomials. The second part discusses the theory of Bernoulli generators on $\text{R}$^N determining systems of orthogonal polynomials that are extensions of the Meixner polynomials to several variables. Linear actions for these spaces are discussed. Some tensors related to the general Bernoulli generators are considered.     

Orthogonal Polynomials on a Bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice ? and on the shifted lattice ?+1???. We combine both lattices to obtain the bi-lattice ???(?+1???) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients that satisfy a nonlinear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.     

Extremal Polynomials on Discrete Sets

We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15.     

Deformed Heisenberg algebra with reflection and <Emphasis Type="Italic">d</Emphasis>-orthogonal polynomials

This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1. The underlying algebraic framework allowed a systematic derivation of the recurrence relations, difference equation, lowering and rising operators and generating functions which these polynomials satisfy.     

Spectral analysis on planar mixed automorphic forms

We introduce and we carry out some concrete spectral properties of a class of magnetic Schrödinger operators leaving invariant the space of the planar mixed automorphic forms of type (ν,μ) with respect to an equivariant pair (ρ,τ) and given lattice Γ⊂C. The associated polynomials constitute classes of generalized complex polynomials of Hermite type.     

A classification of invariant distributions and convergence of imprecise Markov chains

We analyse the structure of imprecise Markov chains and study their convergence by means of accessibility relations. We first identify the sets of states, so-called minimal permanent classes, that are the minimal sets capable of containing and preserving the whole probability mass of the chain. These classes generalise the essential classes known from the classical theory. We then define a class of extremal imprecise invariant distributions and show that they are uniquely determined by the values of the upper probability on minimal permanent classes. Moreover, we give conditions for unique convergence to these extremal invariant distributions.     

On a result of roy and Gnanadesikan concerning multivariete variance components

Summary Roy and Gnanadesikan [5] showed that inference for a general multivariate variance components model may be carried out using the standard multivariateF distribution under certain condtions. It is shown in this note that the theory of zonal polynomials, and their extension by the author to invariant polynomials in two matrix arguments, provide a concise approach to the derivation of these conditions. Relevant distributions are also derived for the general case. CSIRO     

Some Combinatorics of the Hypergeometric Series

Symmetry formulas for the classical hypergeometric series ₂F₁ are proved combinatorially. The idea of the proofs is to find weighted combinatorial structures which form models for each side of the formula and to show how to go from the first to the second model by a ‘weak isomorphism’ (i.e. a sequence of isomorphisms, regroupings and degroupings of structures). This is then applied to the four ₂F₁-families (Meixner, Krawtchouk, Meixner-Pollaczek and Jacobi) of hypergeometric orthogonal polynomials. We give three ‘weakly isomorphic’ models for each family and prove in a completely combinatorial way the 3-terms recurrences for these polynomials.     

Stochastic processes and special functions: On the probabilistic origin of some positive kernels associated with classical orthogonal polynomials

We describe and investigate a class of Markovian models based on a form of “dynamic occupancy problem” originating in statistical mechanics. The most fundamental of these gives rise to a transition-probability matrix over (N + 1) discrete states, which proves to have the Hahn polynomials as eigenvectors. The structure of this matrix, which is a convolution of two negative hypergeometric distributions, leads to a factorization into finite-difference sumoperators having forms analogous to the Erdelyi-Kober operators for the continuous variable. These make possible the exact solution of the corresponding eigenvalue problem and hence the spectral representation of the transition matrix. By taking suitable limits, further families of Markov processes can be generated having other classical polynomials as eigenvectors; these, like the polynomials, inherit their properties from the original Hahn system. The Meixner, Jacobi and Laguerre systems arise in this way, having their origin in variants of the basic model. In the last of these cases, the spectral resolution of the continuous transition kernel proves to be identical with Erdelyi's (1938) bilinear formula, which is thus both generalized and given a physical interpretation. Various symmetry and “duality” properties are explored and a number of interesting formulas are obtained as by-products. The use of statistical models to generate kernels, which are thereby guaranteed to be both positive and positive definite, appears to be mathematically fruitful, while the models themselves seem likely to have application to a variety of topics in applied probability.     

Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

Josef Meixner: His life and his orthogonal polynomials

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner’s 1934 paper introducing the Meixner and Meixner–Pollaczek polynomials is discussed in detail. Truksa’s forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner’s 1934 paper.     

Matrices with prescribed rows and invariant factors

In this paper, we solve the problem of the existence of an n × n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed. The solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.