Exact solutions to a class of differential equation and some new mathematical properties for the universal associated-Legendre polynomials

The exact solutions to a class of differential equation are studied. Some special cases are discussed for the central potentials, single ring-shaped potential and the angular Teukolsky equation. A new expression to the associated Legendre polynomials is found. Some new properties of the universal associated-Legendre polynomials (UALPs) including the generating function, Rodrigues’ formula, parity, some special values and the recurrence relations are presented. A new different Rodrigues’ formula of associated Legendre polynomials is also obtained.     

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.     

Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.      

Connections between Romanovski and other polynomials

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.     

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.     

Generalized Hermite polynomials

Hermite polynomials of several variables are defined by a generalization of the Rodrigues formula for ordinary Hermite polynomials. Several properties are derived, including the differential equation satisfied by the polynomials and their explicit expression. An application is given.     

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.     

Rodrigues type formula for orthogonal polynomials on the unit ball

For a class of weight functions invariant under reflection groups on the unit ball, a family of orthogonal polynomials is defined via a Rodrigues type formula using the Dunkl operators. Their properties and their relation with several other bases are explored.

     

Rodrigues Formulas for the Macdonald Polynomials

We present formulas of Rodrigues type giving the Macdonald polynomials for arbitrary partitionsλthrough the repeated application of creation operatorsB_k,k=1, …, ℓ(λ) on the constant 1. Three expressions for the creation operators are derived one from the other. When the last of these expressions is used, the associated Rodrigues formula readily implies the integrality of the (q, t)-Kostka coefficients. The proofs given in this paper rely on the connection between affine Hecke algebras and Macdonald polynomials.     

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest.The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.     

Hermite Matrix-Valued Functions Associated to Matrix Differential Equations

Some sequences of matrix polynomials have been introduced recently as solutions of certain second-order differential equations, which can be seen as appropriate generalizations, to the matrix setting, of classical orthogonal polynomials. In this paper, we consider families (in a complex parameter) of matrix-valued special functions of Hermite type, which arise as natural extensions of the aforementioned matrix polynomials of the same type. We show that such families are solutions of corresponding differential equations and enjoy several structural properties. In particular, they satisfy a Rodrigues formula expressed in terms of the Weyl fractional calculus. We also show that, unlike the scalar case, a second-order differential operator having such a family as a set of joint eigenfunctions need not be unique.     

The Hermite–Pade Approximations of Generalized Hypergeometric Series in Two Variables

We give examples of well-posed problems of joint Hermite–Pade approximations of series in two variables. We find Rodrigues formulas and integral representations for solutions. We also study the limit distribution of zeros of the corresponding polynomials. Constructions are based, on the one hand, on the classical Appel polynomials orthogonal in a triangle and, on the other hand, on various ways of proving Apery's theorem about irrationality of the number (3).     

Polynomials associated with a functional product of the Hermite type

We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd.     

The Dω—classical orthogonal polynomials

This is an expository paper; it aims to give an essentially self-contained overview of discrete classical polynomials from their characterizations by Hahn’s property and a Rodrigues’ formula which allows us to construct it. The integral representations of corresponding forms are given.     

Rodrigues solutions of the dirac equation for shape-invariant potentials: Master function approach

We show that a Schrödinger-like differential equation for the upper spinor component, derived from the Dirac equation for a charged spinor in spherically symmetric electromagnetic potentials, can be transformed into the Schrödinger equation with some shape-invariant potentials. By choosing different electrostatic potentials and relativistic energies and also introducing new functions and changing the variables, we show that this equation transforms into the differential equations in mathematical physics. We solve these equations using the master function approach and write the spinor wave functions in terms of Rodrigues polynomials associated with these differential equations.     

Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).     

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

   Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).     

The rodrigues type representations for a certain class of special functions

Summary An attempt is made here to present a systematic introduction to and several applications of a certain method of obtaining Rodrigues type representations for a fairly wide variety of sequences of special functions. The main results, contained in Theorems2 and3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for instance, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but also to such other special functions as the Bessel function and a certain class of generalized hypergeometric functions. Entrata in Redazione il 25 giugno 1977. This work was partially supported by the National Research Council of Canada under grants A-7353 and A-4027. For a preliminary report of this paper see Notices Amer. Math. Soc.,24 (1977), p. A-238, Abstract no. 77T-B43.