Last edited by Mizahn
Monday, May 18, 2020 | History

4 edition of q-difference operators, orthogonal polynomials, and symmetric expansions found in the catalog.

# q-difference operators, orthogonal polynomials, and symmetric expansions

## by Douglas Bowman

Written in English

Subjects:
• q-series,
• Difference operators,
• Hypergeometric functions,
• Orthogonal polynomials

• Edition Notes

Classifications The Physical Object Statement Douglas Bowman Series Memoirs of the American Mathematical Society -- no. 757 LC Classifications QA3 .A57 no.757 Pagination ix, 56 p. ; Number of Pages 56 Open Library OL15360906M ISBN 10 082182774X LC Control Number 2002025581

In this paper, we obtain a necessary and sufficient condition on a linear operator J, defined on polynomials, and a d-symmetric d-orthogonal polynomial set {Pn}n≥0 such that {JPn}n≥0 is also d. Zeros. If the measure dα is supported on an interval [a, b], all the zeros of P n lie in [a, b].Moreover, the zeros have the following interlacing property: if m orthogonal polynomials. The Macdonald polynomials are orthogonal polynomials in several variables, depending on the choice of an affine root system.

A generalized Fourier series is a series expansion of a function based on a system of orthogonal polynomials. By using this orthogonality, a piecewise continuous function $${f\left(x \right)}$$ can be expressed in the form of generalized Fourier series expansion. Abstract In this work we apply a q-ladder operator approach to orthogonal polynomials arising from a class of indeterminate moment problems. We derive general representation of first and second order q-difference operators and we study the solution basis of the corresponding second order q-difference equations and its properties.

Orthogonal polynomials and di usions operators D. Bakry, S. Orevkov, M. Zani y Septem Abstract Generalizing the work of [5, 41], we give a general solution to the following prob-lem: describe the triplets (;g;) where g= (gij(x)) is the (co)metric associated to the symmetric second order di erential operator L(f) = 1 ˆ P ij @ i(g. Orthogonal polynomials in Statistics The polynomials commonly used as orthogonal contrasts for quantitative factors are discrtete analogues of Legendre polynomials. One way to understand them is to consider the discretization of the inner product of L2([a,b]): hf,gi = X i=0 t− 1 f(x i)g(x i) where x i is an increasing sequence of points in [a File Size: KB.

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### Q-difference operators, orthogonal polynomials, and symmetric expansions by Douglas Bowman Download PDF EPUB FB2

In special cases these symmetric polynomials reduce to well-known classes of orthogonal polynomials. A number of basic properties of these polynomials follow from our approach.

This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. We also find expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials. This Cited by: We explore ramifications and extensions of a $$q$$-difference operator method first used by L.J.

Rogers for deriving relationships between special functions involving certain fundamental $$q$$-symmetric polynomials. In special cases these symmetric polynomials reduce to well-known classes of orthogonal q-difference operators. Get this from a library. Q-difference operators, orthogonal polynomials, and symmetric expansions.

[Douglas Bowman]. q-difference operators, orthogonal polynomials, and symmetric expansions / Douglas Bowman Article in Memoirs of the American Mathematical Society () September with 25 Reads. The q -Legendre polynomials are defined by the Rodrigues formula to enable an easy orthogonality relation.

q -Legendre polynomials have been and symmetric expansions book before, but these do not have the same orthogonality range in the limit as ordinary Legendre polynomials. We also find q -difference equations for these by: 7.

Symmetric Functions and Orthogonal Polynomials I. MacDonald One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials has long been known to be connected to combinatorics, representation theory, and other branches of mathematics. There are a number of intriguing connections between Painlevé equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these.

Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlevé equations appear in mathematical analysis very by: k is a constant times the orthogonal polynomial p k related to α See Szeg¨o We have deﬁned orthogonality relative to an inner product given by a Riemann–Stieltjes integral but, more generally, orthogonal polynomials can and symmetric expansions book deﬁned relative to a linear functional L such that L(λk) = µ k Two polynomials p and q are said to be orthogonal File Size: KB.

Orthogonal polynomials We start with Deﬂnition 1. A sequence of polynomials fpn(x)g1 n=0 with degree[pn(x)] = n for each n is called orthogonal with respect to the weight function w(x) on the interval (a;b) with a File Size: KB.

[−1,1]. The method uses the discrete orthogonal polynomial least squares (DOP-LS) ap-proximation based on the super Gaussian weight function, which is both smoothly con-nected to zero at ±1 and equals one in nearly the entire domain. As a result, the method has fast decaying expansion coefﬁcients and also successfully suppresses Runge oscil-Cited by: Orthogonal Polynomials 75 where the Yij are analytic functions on C \ R, and solve for such matrices the following matrix-valued Riemann–Hilbert problem: 1.

for all x ∈ R Y +(x) = Y −(x) 1 w(x) 0 1 where Y +, resp. Y −, is the limit of Y(z) as z tends to x from the upper, resp. lower half plane, andFile Size: KB.

We establish one general q-exponential give one general q-difference onship between q-exponential operator and q-difference equation are q-difference equation to set up a q-polynomials generating functions for Hn are obtained. Get this from a library.

Q-difference operators, orthogonal polynomials, and symmetric expansions. [Douglas Bowman] -- Introduction and preliminaries New results and connections with current research Vector operator identities and simple applications Bibliography.

equations using orthogonal polynomials. For example consider the Schr odinger equation, 1 2 @2 @x2 + V(x) = E The di erential operator acting on the function is linear, H^ = 1 2 @2 @x2 + V(x) H^ (f(x) + g(x)) = Hf^ (x) + Hg^ (x) Spencer Rosenfeld Orthogonal Polynomials October 24 10 / 14File Size: KB.

Orthogonal Functions, Orthogonal Polynomials, and Orthogonal Wavelets series expansions of function Sergey Moiseev Kodofon. Russia. [email protected], [email protected] As you read this worksheet, you should execute the commands in sequence as.

The point here is that if we ﬁnd an orthogonal basis B, we would be able to approximate or decompose a function f by the rule f ∼= X g∈B hf,gi hg,gi g.

The above is an equality if f ∈ span(B), that is, f is a linear combination of some functions in B. Otherwise, it is an orthogonal projection of f onto span(B). 2 Orthogonal PolynomialsFile Size: 79KB. Reviews: This is the first detailed systematic treatment of (a) the asymptotic behaviour of orthogonal polynomials, by various methods, with applications, in particular, to the ‘classical' polynomials of Legendre, Jacobi, Laguerre and Hermite; (b) a detailed study of expansions in series of orthogonal polynomials, regarding convergence and summability; (c) a detailed study of orthogonal.

Transfer Operators in the Context of Orthogonal Polynomials Sina F. Straub symmetric. We construct an orthogonal polynomial sequence based on a quadratic trans-formation which is given by the second order Chebyshev polynomial of the rst kind, T 2 p xq 2x2 1.

We can show that the constructed orthogonal polynomial sequence. Theorem (a) Orthogonal polynomials always exist. (b) The orthogonal polynomial of a ﬁxed degree is unique up to scaling.

(c) A polynomial p 6= 0 is an orthogonal polynomial if and only if hp,qi = 0 for any polynomial q with degq polynomial p 6= 0 is an orthogonal polynomial if and only if hp,xki = 0 for any 0 ≤ k File Size: KB. The Askey–Wilson polynomials are the most general orthogonal polynomials, which are eigenfunctions of a second order q-difference operator.

I survey recent results aiming at constructing all orthogonal polynomials which are eigenfunctions of a q-difference operator of an arbitrary order, by means of the lattice Darboux : Luc Haine. D. BowmanA general Heine transformation and symmetric polynomials of Rogers.

D. Bowman, q-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions, submitted. Google Scholar. 9. L. CarlitzSome polynomials related to theta by: 7.these finite orthogonal polynomials.

lf the points are not eaually spaced, the work of this thesis does not apply, (unless the data may be grouped in some manner so that the resulting points are equally spaced.) To work with these polynomials, some knowledge of the calculus of finite differences is needed.

Here, the fac­.Symmetry in orthogonal polynomials also appears when the domain is a symmetric shape and the weight function is invariant under a group generated by reflections: for example orthogonal polynomials on a regular hexagon find an application in wave-front analysis for hexagonal mirror segments in large astronomical telescopes.