The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomial...

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the measure |x|γ(1 − x2)1/2dx is derived which is based on a “reversing property” of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalizati...

In the present paper, the author applies some of his earlier results which extend the well-known Hille-Hardy formula for the Laguerre polynomials to certain classes of generalized hypergeometric polynomials in order to derive various generalizations of a bilinear generating function for the Jacobi polynomials proved recently by Carlitz. The corresponding results for the polynomials of Legendre,...

We look for differential equations of the form ∞ ∑ i=0 ci(x)y (x) = 0, where the coefficients {ci(x)} ∞ i=0 are continuous functions on the real line and where {ci(x)} ∞ i=1 are independent of n, for the generalized Jacobi polynomials { P n (x) } ∞ n=0 and for generalized Laguerre polynomials { L n (x) } ∞ n=0 which are orthogonal with respect to an inner product of Sobolev type. We use a metho...

We look for differential equations of the form ∞ ∑ i=0 ci(x)y (x) = λny(x), where the coefficients {ci(x)} ∞ i=0 do not depend on n, for the generalized Jacobi polynomials { P n (x) } ∞ n=0 found by T.H. Koornwinder in 1984 and for generalized Laguerre polynomials { L n (x) } ∞ n=0 which are orthogonal with respect to an inner product of Sobolev type. We introduce a method which makes use of co...

We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Padé polynomials) of type II. These polynomials can be written as a Jacobi-Piñeiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by T.H. Koornwinder. Here we need to introduce Jacobi and JacobiPiñeiro polynomials with complex parameters. Som...

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the ‘holes’ in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two differe...

In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck’s work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presente...

We study the properties of three families of exponential Riordan arrays related to the Stirling numbers of the first and second kind. We relate these exponential Riordan arrays to the coefficients of families of orthogonal polynomials. We calculate the Hankel transforms of the moments of these orthogonal polynomials. We show that the Jacobi coefficients of two of the matrices studied satisfy ge...

In this note we consider the D optimal design problem for the heteroscedastic polynomial regression model Karlin and Studden a found explicit solutions for three types of e ciency functions We introduce two new functions to model the heteroscedastic structure for which the D optimal designs can also be found explicitly The optimal designs have equal masses at the roots of generalized Bessel pol...