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2 results

  • q-Integral and Moment Representations for q-Orthogonal Polynomials

  • Mourad E. H. Ismail, Dennis Stanton
  • Journal:
    Canadian Journal of Mathematics / Volume 54 / Issue 4 / 01 August 2002
    Published online by Cambridge University Press:
    20 November 2018, pp. 709-735
    Print publication:
    01 August 2002
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  • We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\text{ }{{\varepsilon }_{q}}$.

  • Two Families of Associated Wilson Polynomials

  • M. E. H. Ismail, J. Letessier, G. Valent, J. Wimp
  • Journal:
    Canadian Journal of Mathematics / Volume 42 / Issue 4 / 01 August 1990
    Published online by Cambridge University Press:
    20 November 2018, pp. 659-695
    Print publication:
    01 August 1990
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  • Two families of associated Wilson polynomials are introduced. Both families are birth and death process polynomials, satisfying the same recurrence relation but having different initial conditions. Contiguous relations for generalized hypergeometric functions of the type 7F6 are derived and used to find explicit representations for the polynomials and to compute the corresponding continued fractions. The absolutely continuous components of the orthogonality measures of both families are computed. Generating functions are also given.