Stochastic Processes and Orthogonal Polynomials

Abstract

Library of Congress Cataloging-in-Publication Data Schoutens, Wim. Stochastic processes and orthogonal polynomials / Wim Schoutens. p. cm.-(Lecture notes in statistics; 146) Includes bibliographical references and index.

Modeling uncertainty in flow simulations via generalized polynomial chaos

“…More recent work has shown that an important class of orthogonal polynomials belongs to the Askey scheme of the hypergeometric polynomials [17]. In this section, we briefly review the theory of hypergeometric orthogonal polynomials; we adopt the notation of [21,22].…”

“…The Askey scheme, which is represented as a tree structure in figure 1 (following [22]), classifies the hypergeometric orthogonal polynomials and indicates the limit relations between them. The 'tree' starts with the Wilson polynomials and the Racah polynomials on the top.…”

“…For a detailed account of the limit relations of Askey scheme, the interested reader should consult [21] and [22].…”

Conditional moments of q-Meixner processes

“…Since corr(ξ k−1 , ξ k ) = t k−1 /t k = 0, ±1, the assumption [33, (10)] holds true, too. Finally, notice that Weso lowski's α k = 1, and his (27) we see that D = 0 if and only if C = b. Thus [33, (11)] hold true.…”

“…We can solve this equation for E(X 3 u |F ≤s ), as (27) implies that C = b. Using (24) and (19), the answer simplifies to…”