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Richard Askey

Bio: Richard Askey is an academic researcher from University of Wisconsin-Madison. The author has contributed to research in topics: Orthogonal polynomials & Classical orthogonal polynomials. The author has an hindex of 41, co-authored 154 publications receiving 8012 citations. Previous affiliations of Richard Askey include Northwestern University & Pennsylvania State University.


Papers
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Book
01 Jun 1975
TL;DR: Asymptotics of Jacobi matrices for a family of fractal measures have been studied in this paper, where the authors focus on numerical results and conjectures on the median of the beta distribution, and study the monotonicity and asymptotic properties of the median as a univariate function of the parameter a.
Abstract: Asymptotics of Jacobi matrices for a family of fractal measures GÖKALP APLAN BILKENT UNIVERSITY, TYRKEY There are many results concerning asymptotics of orthogonal polynomials and Jacobi matrices associated with a measure whose essential support is a finite union of intervals on R. In the case of totally disconnected support, spectral theory of orthogonal polynomials is less complete and numerical evaluations can be seen as a useful source of information. In this talk, we discuss various properties and asymptotics of Jacobi matrices for a special family of fractal measures. We focus on numerical results and conjectures. The talk is based on a joint work with Alexander Goncharov and Ahmet Nihat Şimşek. On the median of the beta distribution DIMITRIOS ASKITIS UNIVERSITY OF COPENHAGEN The median q of the beta distribution is defined implicitly by the equation ∫ q 0 ta−1(1− t)b−1dt = 1 2 ∫ 1 0 ta−1(1− t)b−1dt. We study the monotonicity and asymptotic properties of the median as a univariate function of the parameter a, as well as of the function a log q(a), related to its logarithm. In particular, we find asymptotic expansions for a → 0 and ∞. These are related to the polygamma function and generalised Bernoulli polynomials.

802 citations

Book ChapterDOI
01 Jan 1999

636 citations

Book ChapterDOI
01 Jan 1985
TL;DR: The classical orthogonal polynomials have been defined in this paper, and a number of orthogonality relations for some of the classical polynomial classes have been established.
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.

532 citations

Book
01 Jan 1987

451 citations


Cited by
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Book
01 Jan 1979
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
Abstract: I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions with two parameters VII. Zonal polynomials

8,730 citations

Posted Content
18 Dec 2005
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Abstract: In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

5,648 citations

Journal ArticleDOI
TL;DR: This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error.
Abstract: We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs

4,473 citations

Journal ArticleDOI
TL;DR: The quantum group SU(2)q is discussed in this paper by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose.
Abstract: The quantum group SU(2)q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed

1,555 citations

Posted Content
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
Abstract: We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

1,459 citations