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2 results

  • 1 - Spectral Representation of Discrete-Time Birth–Death Chains

  • Manuel Domínguez de la Iglesia, Universidad Nacional Autónoma de México
  • Book:
    Orthogonal Polynomials in the Spectral Analysis of Markov Processes
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 1-56

  • Summary

    This chapter gives an introduction to orthogonal polynomials. It also includes the concept of the Stieltjes transform and some of its properties, which will play a very important role in the spectral analysis of discrete-time birth–death chains and birth–death processes. A section on the spectral theorem for orthogonal polynomials (or Favard’s theorem) will give insights into the relationship between tridiagonal Jacobi matrices and spectral probability measures. The chapter then focuses then on the classical families of orthogonal polynomials, of both continuous and discrete variables. These families are characterized as eigenfunctions of second-order differentials or difference operators of hypergeometric type solving certain Sturm–Liouville problems. These classical families are part of the so-called Askey scheme.

  • Classical Orthogonal Polynomials as Moments

  • Mourad E. H. Ismail, Dennis Stanton
  • Journal:
    Canadian Journal of Mathematics / Volume 49 / Issue 3 / 01 June 1997
    Published online by Cambridge University Press:
    20 November 2018, pp. 520-542
    Print publication:
    01 June 1997
    • Article
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  • We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.