scholarly journals Stochastic processes and special functions: On the probabilistic origin of some positive kernels associated with classical orthogonal polynomials

1977 ◽  
Vol 61 (1) ◽  
pp. 262-291 ◽  
Author(s):  
R.D Cooper ◽  
M.R Hoare ◽  
Mizan Rahman
Keyword(s):  
Stochastic Processes ◽  
Orthogonal Polynomials ◽  
Special Functions ◽  
Classical Orthogonal Polynomials ◽  
Positive Kernels

Orthogonal Polynomials in the Spectral Analysis of Markov Processes

2021 ◽  
Author(s):  
Manuel Domínguez de la Iglesia
Keyword(s):  
Stochastic Processes ◽  
Orthogonal Polynomials ◽  
Markov Processes ◽  
Diffusion Processes ◽  
Special Functions ◽  
Extensive Literature ◽  
Associated Operators ◽  
And Diffusion ◽  
Selection Of ◽  
Birth Death

In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of associated operators. In the decades since, many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains, birth-death processes and diffusion processes. It brings together the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter, it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability.

scholarly journals Some generating-function equivalences

1975 ◽  
Vol 16 (1) ◽  
pp. 34-39 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Generating Relations ◽  
Sequence Of Functions

A generalization is given of a theorem of F. Brafman [1] on the equivalence of generating relations for a certain sequence of functions. The main result, contained in Theorem 2 below, may be applied to several special functions including the classical orthogonal polynomials such as Hermite, Jacobi (and, of course, Legendre and ultraspherical), and Laguerre polynomials.

scholarly journals On Koornwinder classical orthogonal polynomials in two variables

2012 ◽  
Vol 236 (15) ◽  
pp. 3817-3826 ◽  
Author(s):  
Lidia Fernández ◽  
Teresa E. Pérez ◽  
Miguel A. Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Classical Orthogonal Polynomials
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