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

/pdf/the-wiener-askey-polynomial-chaos-for-stochastic-4ne2oljrfc.pdf

The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

Modeling uncertainty in flow simulations via generalized polynomial chaos

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecome large. When all but ﬁnitely many, say r, eigenvalues of the covariance matrix arethe same, the dependence of the limiting distribution of the largest eigenvalue of the samplecovariance matrix on those distinguished r eigenvalues of the covariance matrix is completelycharacterized in terms of an inﬁnite sequence of new distribution functions that generalizethe Tracy-Widom distributions of the random matrix theory. Especially a phase transitionphenomena is observed. Our results also apply to a last passage percolation model and aqueuing model. 1 Introduction Consider M independent, identically distributed samples y 1 ,...,~y M , all of which are N ×1 columnvectors. We further assume that the sample vectors ~y k are Gaussian with mean µ and covarianceΣ, where Σ is a ﬁxed N ×N positive matrix; the density of a sample ~y isp(~y) =1(2π)

/pdf/phase-transition-of-the-largest-eigenvalue-for-nonnull-2kdsrttosc.pdf

Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.

A "missing" family of classical orthogonal polynomials

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say $r$, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished $r$ eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the random matrix theory. Especially a phase transition phenomena is observed. Our results also apply to a last passage percolation model and a queuing model.

Phase transition of the largest eigenvalue for non-null complex sample covariance matrices

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.

/pdf/the-askey-scheme-of-hypergeometric-orthogonal-polynomials-46afheftah.pdf

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Definitions and Miscellaneous Formulas- Classical orthogonal polynomials- Orthogonal Polynomial Solutions of Differential Equations- Orthogonal Polynomial Solutions of Real Difference Equations- Orthogonal Polynomial Solutions of Complex Difference Equations- Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations- Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations- Hypergeometric Orthogonal Polynomials- Polynomial Solutions of Eigenvalue Problems- Classical q-orthogonal polynomials- Orthogonal Polynomial Solutions of q-Difference Equations- Orthogonal Polynomial Solutions in q?x of q-Difference Equations- Orthogonal Polynomial Solutions in q?x+uqx of Real

Hypergeometric Orthogonal Polynomials and Their q-Analogues

For H. Exton's q-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson's q-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskii. As a specialization we get q-cosines and q-sines which admit q-analogues of the Fourier-cosine and Fourier-sine transforms

/pdf/on-q-analogues-of-the-fourier-and-hankel-transforms-etgwc4oxez.pdf

On q-analogues of the Fourier and Hankel transforms

On the Zeros of the Hahn-Exton q-Bessel Function and Associated q-Lommel Polynomials

/pdf/the-hahn-exton-q-bessel-function-10r9tsd0s3.pdf

René F. Swarttouw

Papers

The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Hypergeometric Orthogonal Polynomials and Their q-Analogues

On q-analogues of the Fourier and Hankel transforms

On the Zeros of the Hahn-Exton q-Bessel Function and Associated q-Lommel Polynomials

The Hahn-Exton q-Bessel Function