Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials

We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials p_k(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑ_m(x) p_n(x;q), where ϑ_m(x) is x^mor (x;q)_m, and the expansion of the Hahn-Exton q-Bessel function in the little q-Jacobi polynomials are discussed in detail. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation,Factorization and Solutions

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.     

Orthogonal polynomials—centroid of their zeroes

Let c _n,k (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P _n (x). The centroid of these zeroes: $s_n=\frac1n \sum\limits^n_{k=1}c_{n,k}$ controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s _n , like bounds, inequalities, parameters dependence, ..., from the links between s _n , the coefficients of the expansion of P _n (x), and the coefficients β _n , γ _n in the basic recurrence relation satisfied by P _n (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal 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.     

Exponential of Gel'fand lattices and irreducible representations of U(n)

A finite difference equation defines the exponential of a square tableau, extension of the usual Gel'fand pattern. The application to the group U(n) gives explicitly the Gel'fand states for n = 4 and the matrix elements for n = 3.     

Rediscovering the contributions of Rodrigues on the representation of special functions

Following the pressure of Hermite in 1865 and Heine in 1868, the name of Rodrigues was definitively associated to his famous representation given in 1815. In this paper, we precise the contribution of Rodrigues, proving that his formula not only generates the Legendre polynomials, but also special Gegenbauer polynomials and even the corresponding second kind functions, which are never mentioned. The links with Hildebrandt polynomials and with the Bell polynomials are also briefly examined.     

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product with and (x)is a weight function.We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to (x) and we find the recurrence relation verified by such a family. If the weight is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.     

Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.     

Bivariate second-order linear partial differential equations and orthogonal polynomial solutions

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed.