Some results on co-recursive associated Meixner and Charlier polynomials

An explicit representation of the co-recursive associated Meixner polynomials is given in terms of hypergeometric functions. This representation allows to derive a generating function, the Stieltjes transform of the orthogonality measure and the fourth-order difference equation verified by these polynomials. Special attention is given to some simple limiting cases ocurring in the solution of the Chapman-Kolmogorov equation of linear birth and death processes.     

Two-dimensional orthogonal polynomials,their associated sets and the co-recursive sets

This paper deals with the general theory of sets of polynomials verifying a (d+1)-order recurrence. The cased=2 is specially carried out. First, we introduce the notion of associated set of a set of monic polynomials. A general formula for successive associated polynomials is given. The co-recursive sets of a two-dimensional orthogonal set are introduced. We calculate the corresponding formal Stieltjes functions.Finally, we determine the self-associated two-dimensional orthogonal sets and we show they are classical two-dimensional orthogonal sets, that is to say, their set of derivatives is also a two-dimensional orthogonal set.     

On sieved orthogonal polynomials. VIII. Sieved associated Pollaczek polynomials

We study a general orthogonal polynomial set which includes the sieved associated ultraspherical and the sieved Pollaczek polynomials. This we get by letting q approach a root of unity in the recurrence relation and the generating functions of the associated q-ultraspherical and the Pollaczek polynomials. We find the weight functions with respect to which these polynomials are orthogonal and determine the asymptotic behavior of these polynomials on and off their interval of orthogonality.     

On biorthogonal systems associated to orthogonal polynomials

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.     

On associated polynomials and decay rates for birth-death processes

We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two measures, and their moments. As an application, we analyze the relation between two decay rates connected with a birth-death process.     

On hyperbolicity cones associated with elementary symmetric polynomials

Elementary symmetric polynomials can be thought of as derivative polynomials of . Their associated hyperbolicity cones give a natural sequence of relaxations for . We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative characterization of these cones, and give an algebraic characterization for one particular dual cone associated with together with its self-concordant barrier functional.     

On Eisenstein polynomials and zeta polynomials

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.     

On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method

In this paper we explore two sets of polynomials, the orthogonal polynomials and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix C A while the roots of the residual polynomials, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (C A)^?1. When C A is selfadjoint positive definite, this distinction is minimal, but for the indefinite or nonselfadjoint case this distinction becomes important. We use these two sets of roots to form possibly nonconvex regions in the complex plane that describe the spectrum of C A.     

On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P ([0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.     

Polynomial evaluation and associated polynomials

We show a connection between the Clenshaw algorithm for evaluating a polynomial , expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial given in monomial (!) representation can be evaluated for using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients. Received January 2, 1995 / Revised version received April 9, 1997     

On difference polynomials and hereditarily irreducible polynomials

A difference polynomial is one of the form P(x, y) = p(x) ? q(y). Another proof is given of the fact that every difference polynomial has a connected zero set, and this theorem is applied to give an irreducibility criterion for difference polynomials. Some earlier problems about hereditarily irreducible polynomials (HIPs) are solved. For example, P(x, y) is called a HIP (two-variable case) if P(a(x), b(y)) is always irreducible, and it is shown that such two-variable HIPs actually exist.     

On the fourth-order difference equation for the associated Meixner polynomials

Three equivalent forms of the fourth-order difference equation obeyed by the associated Meixner polynomials (with a nonnegative real association parameter) are derived from a refinement of a recent result due to Letessier et al. (1996).     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001