Open AccessPosted Content
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
Roelof Koekoek,René F. Swarttouw +1 more
Reads0
Chats0
TLDR
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.read more
Citations
More filters
Journal ArticleDOI
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
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.
Journal ArticleDOI
Modeling uncertainty in flow simulations via generalized polynomial chaos
TL;DR: In this paper, the authors present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations, which is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space.
Journal ArticleDOI
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
TL;DR: In this paper, the limiting distribution of the largest eigenvalue of a complex Gaussian covariance matrix was studied in terms of a sequence of new distribution functions that generalize the Tracy-Widom distribution of random matrix theory.
Journal ArticleDOI
A "missing" family of classical orthogonal polynomials
Luc Vinet,Alexei Zhedanov +1 more
TL;DR: In this article, a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type was studied.
Posted Content
Phase transition of the largest eigenvalue for non-null complex sample covariance matrices
TL;DR: In this paper, the limiting distribution of the largest eigenvalue of a complex Gaussian covariance matrix when both the number of samples and variables in each sample become large is studied.
References
More filters
Reference BookDOI
Asymptotics and Special Functions
TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
Book
Basic Hypergeometric Series
George Gasper,Mizan Rahman +1 more
TL;DR: In this article, the Askey-Wilson q-beta integral and some associated formulas were used to generate bilinear generating functions for basic orthogonal polynomials.