Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Weights whose biorthogonal polynomials admit a Rodrigues formula

Let α>0 and ψ(x)=xα. Let w be a non-negative integrable function on an interval I. Let Pn be a polynomial of degree n determined by the biorthogonality conditions

     

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r_k}_k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r_k}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.     

Orthogonal polynomials and differential equations in neutron-transport and radiative-transfer theories

There is a set of orthogonal polynomials {g_n(x)} which plays a relevant role in the treatment of the case of anisotropic scattering in neutron-transport and radiative-transfer theories. They appear also in the spherical harmonics treatment of the isotropic scattering. These polynomials are orthogonal with respect to a weight function which is continuous in the interval [−1, + 1] and has a finite number of symmetric Dirac masses. Although some other structural properties of these polynomials (e.g., the three-term recurrence relation) as well as some properties of their zeros have been published, much more need to be known. In particular, neither the second-order differential equation nor the density of zeros (i.e., the number of zeros per unit of interval) of the polynomial g_n(x) have been found. Here we obtain the second-order differential equation in the case that these polynomials are hypergeometric, so leaving open the general case. Furthermore, the exact expressions of the moments around the origin of the density of zeros of g_n(x) are given in the general case. The asymptotic density of zeros is also pointed out. Finally, these polynomials are shown to belong to the Nevai's class.     

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83-109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88-112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261-280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359-392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.     

Differential coefficients of orthogonal matrix polynomials

We find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given.     

Solvability of matrix equations in rings of quasi-diagonal matrices and similarity of matrix polynomials

A certain standard form is found for a complex matrix with respect to equivalent transformations by quasi-diagonal matrices. The solvability of certain matrix equations in the rings of quasi-diagonal matrices is examined using this standard form.     

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.

     

Rakhmanov's theorem for orthogonal matrix polynomials on the unit circle

Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle.     

A characterization of the Pearson system of distributions and the associated orthogonal polynomials

A new derivation of the classical orthogonal polynomials is given by using thew-function which appears in the variance bounds and some properties of the Pearson system of distributions. Also a characterization of the Pearson system of distributions through some conditional moments is obtained by using a result obtained by Johnson (1993) concerning this family.     

A miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations

We find structural formulas for a family (P_n)_n of matrix polynomials of arbitrary size orthogonal with respect to the weight matrix e^−t2e^Ate^A∗t, where A is certain nilpotent matrix. It turns out that this family is a paradigmatic example of the many new phenomena that show the big differences between scalar and matrix orthogonality. Surprisingly, the polynomials P_n, n≥0, form a commuting family. This commuting property is a genuine and miraculous matrix setting because, in general, the coefficients of P_n do not commute with those of P_m, n≠m.     

Orthogonal polynomials for exponential weights on

Let I=[0,d), where d is finite or infinite. Let , where and Q is continuous and increasing on I, with limit ∞ at d. We study the orthonormal polynomials associated with the weight , obtaining bounds on the orthonormal polynomials, zeros, and Christoffel functions. In addition, we obtain restricted range inequalities.     

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

The properties of matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators.A new, direct proof is given to the conjecture of Durán and Grünbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type.Commutative classes of orthogonal polynomials (corresponding to weights that are self-adjoint but not positive semidefinite) are found, which satisfy all the properties usually associated to orthogonal polynomials, and are not of scalar type.     

Second order partial differential equations for gradients of orthogonal polynomials in two variables

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.     

Orthogonal polynomials and analytic functions associated to positive definite matrices

For a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation An of size n×n. For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems, respectively.     

Polynomial extensions of distributions and their applications in actuarial and financial modeling

The paper deals with orthogonal polynomials as a useful technique which can be attracted to actuarial and financial modeling. We use Pearson’s differential equation as a way for orthogonal polynomials construction and solution. The generalized Rodrigues formula is used for this goal. Deriving the weight function of the differential equation, we use it as a basic distribution density of variables like financial asset returns or insurance claim sizes. In this general setting, we derive explicit formulas for option prices as well as for insurance premiums. The numerical analysis shows that our new models provide a better fit than some previous actuarial and financial models.     

A Rodrigues-type formula for Gegenbauer matrix polynomials

This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given.     

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form

W(t)=t^αe^-te^Att^Bt^B*e^A*t,