Showing papers on "Discrete orthogonal polynomials published in 1985"

Classical orthogonal polynomials

Abstract: There have been a number of definitions of the classical orthogonal polynomials, but each definition has left out some important orthogonal polynomials which have enough nice properties to justify including them in the category of classical orthogonal polynomials. We summarize some of the previous work on classical orthogonal polynomials, state our definition, and give a few new orthogonality relations for some of the classical orthogonal polynomials.

Where Does the Sup Norm of a Weighted Polynomial Live? (A Generalization of Incomplete Polynomials)

Abstract: A characterization is given of the sets supporting the uniform norms of weighted polynomials (w(x))nPn(x), where Pn is any polynomial of degree at most n. The (closed) support r, of w(x) may be bounded or unbounded; of special interest is the case when w(x) has a nonempty zero set Z. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of r~ - Z. One main result of this paper states that there is a unique compact set (in- dependent of n and P,) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights (w(x))" is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.

A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials

Abstract: The polynomial sets {Y"(x; k)} and { Z"(x; &)}, discussed by Joseph D. E. Konhauser, are biorthogonal over the interval (0, oo) with respect to the weight function x a e~ x , where a > — 1 and A: is a positive integer. The object of the present note is to develop a fairly elementary method of proving a general multilinear generating function which, upon suitable specializations, yields a number of interesting results including, for example , a multivariable hypergeometric generating function for the multiple sum: 1. Introduction. Joseph D. E. Konhauser ([5]; see also [4]) introduced two interesting classes of polynomials: Y*(x\ k) a polynomial in JC, and Z"(JC; k) a polynomial in JC*, α >-1 and k = 1,2,3, For fc = 1, these polynomials reduce to the classical Laguerre polynomials L^ α) (x), and for k = 2 they were encountered earlier by Spencer and Fano [8] in the study of the penetration of gamma rays through matter and were discussed subsequently by Preiser [7]. Also [5, p. 303]

New inequalities for polynomials

Abstract: Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Ricsz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.

The Hahn and Meixner polynomials of an imaginary argument and some of their applications

Abstract: The Hahn and Meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter. This leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure. The Meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the Pollaczek polynomials. The orthogonality relation for the Hahn polynomials with respect to a continuous measure is apparently new. A close connection between the Hahn polynomials of an imaginary argument and representations of the Lorentz group SO(3,1) is considered.

Continuous Hahn polynomials

Abstract: A slightly more general orthogonality relation for the Hahn polynomials of a continuous variable than the recent one given by Atakishiyev and Suslov, (1985), is given here.

Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Abstract: Consider a system {φ n } of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′>0 almost everywhere. Denoting byk n the leading coefficient ofφ n , a simple new proof is given for E. A. Rakhmanov's important result that lim n→∞,k n /k n+1=1; this result plays a crucial role in extending Szego's theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L 1 to a wider class of orthogonal polynomials.

Additive Complexity and Zeros of Real Polynomials

Abstract: Let $P \in R[ X ]$ be a polynomial of additive complexity k (the additive complexity is the minimal number of $ \pm $ operations needed to compute P over R). It is shown that there exists a constant C (independent of P) such that the number of distinct real zeros of P is $ \leqq C^{k^2 } $.This is an improvement on a result of Borodin and Cook (SIAM J. Comput., 5 (1970), pp. 146–157). This result is then generalized to polynomials in several variables, the number of zeros being replaced by the number of connected components of the zero set.

Asymptotics for orthogonal polynomials defined by a recurrence relation

Abstract: Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.

Characteristic polynomials of organic polymers and periodic structures

Abstract: It is shown that the Frame’s method (also, Le Verrier-Faddeev’s method) for characteristic polynomials of chemical graphs can be extended to periodic graphs and structures. The finite periodic structures are represented by cyclic structures in the Born-von Khnan boundary condition which leads to complex matrices. In this article we demonstrate that our earlier computer program (based on Frame’s method) can be extended to these periodic networks. The characteristic polynomials of several lattices such as polydiacetylenes, one-dimensional triangular, square, and hexagonal lattices are obtained. These polynomials can be obtained with very little computer time using this method. I. INTRODUCTION In recent years characteristic polynomials and related polynomials of graphs and other applications of graph theory to chemistry have been the subjects of a large number of inve~tigations.l-~’ Characteristic polynomials play an important role in several areas of chemistry. These polynomials are structural invariants and are thus useful in coding chemical structures. They are generating functions for dimer statistics on trees (such as Bethe lattices) and thus they play an important role in statistical mechanic^.^^.^,^^ Characteristic polynomials of graphs have applications in quantum chemistry, chemical kinetics, dynamics of oscillatory reactions, etc. They are also useful in estimating the stability of conjugated systems. The present author showed the use of Frame’s method for evaluating characteristic polynomials of graphs containing large numbers of vertices and further developed a computer program based on this meth~d.~~.~~ Krivka, Jericevib, and Trinajstib4’ have recently shown that the Frame’s method outlined in the present author’s article is similar to Le Verrier-Faddeev’s method. Other manifestations of Frame’s method could also be found in the literat~re.~~ The objective of this investigation is to show that the method and computer program

Asymptotic Expansions of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights

Abstract: Letp„(x) = y„x\" + •■ ■ denote the nth polynomial orthonormal with respect to the weight exp(-x^/ß) where ß > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing a„ = t„-\\/in, the expression a„n~1/P has a limit as n -» oo. It is shown that this expression has an asymptotic expansion in terms of negative even powers oí n. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored. Consider the polynomials pn that are orthonormal with respect to the weight function exp(-\\x\\ß/ß) on the real line, where ß is a positive real number. Denoting by y„ the leading coefficient of pn (n > 0) and writing an = y„ i/y„ for n > 1 and an = 0 for n < 0, G. Freud conjectured that / jB-1 Y1/ß W A\"L°-/\"'/'=((/-2)/2) holds for every positive even ß (see [3, Conjecture, p. 5]; his conjecture has a slightly different form, as he considered the weight function |jc|pexp(—|jc|^) rather than the one above). He also entertained the possibility that this conjecture is valid for all positive real ß. In case ß > 0 is even, he proved that if the limit on the left exists then it must have the value on the right-hand side (see [3, Theorem 1 on p. 4]), and he established the conjecture for ß = 2, 4, and 6 (see [3, pp. 5-6]). He accomplished these by extracting information from the formula (2) —=\\ Pn(x)p„-X(x)x-l\\x\\ exp -|x| /ß) dx, Received by the editors October 30, 1983. 1980 Mathematics Subject Classification. Primary 42C05; Secondary 05A15, 05A19, 41A60.

Matrix Elements of Irreducible Representations of $SU(2) \times SU(2)$ and Vector-Valued Orthogonal Polynomials

Abstract: The matrix elements of irreducible representations of $SU(2) \times SU(2)$ in a ${\operatorname{diag}}SU(2) \times SU(2)$-basis are expressed in terms of vector-valued orthogonal polynomials, which generalize the Jacobi polynomials.

On Sieved Orthogonal Polynomials I: Symmetric Pollaczek Analogues

Abstract: Two sieved analogues of the Pollaczek polynomials are introduced and the weight functions for the new polynomials are computed. Various asymptotic and explicit formulas are derived. Generating functions are also included.

The attractive Coulomb potential polynomials

Abstract: TheJ matrix method in quantum mechanics developed by Heller, Reinhardt, and Yamani points to a set of orthogonal polynomials having a nonempty continuous spectrum in addition to an infinite discrete spectrum. Asymptotic methods are used to investigate the spectral properties of these polynomials. We also obtain generating functions for both numerator and denominator polynomials. The corresponding continued fraction is computed and the Stieltjes inversion formula is used to determine the distribution function.

On discrete scattering Hurwitz polynomials

Abstract: Discrete scattering Hurwitz polynomials play for discrete systems a similar role as scattering Hurwitz polynomials do for classical systems. A number of properties of discrete scattering Hurwitz polynomials are presented. These properties are obtained by deriving them from the corresponding properties of scattering Hurwitz polynomials. This is achieved by considering polynomials associated in a proper way to given polynomials and by examining the nature of the corresponding transformations.

Inequalities concerning the derivative of polynomials

Abstract: A simple proof of Bernstein's Theorem on the derivative of polynomials is presented.

Least-squares fitting using orthogonal multinomials

Abstract: Forsythe has given a method for generating basis polynomials in a single variable that are orthogonal with respect to a given inner product. Weisfeld later demonstrated that Forsythe's approach could be extended to polynomials in an arbitrary number of variables. In this paper we sharpen Weisfeld's results and present a method for computing weighted, multinomial, least-squares approximations to given data.