Classical Orthogonal Polynomials of a Discrete Variable

  • Arnold F. Nikiforov
  • Vasilii B. Uvarov
  • Sergei K. Suslov
Part of the Springer Series in Computational Physics book series (SCIENTCOMP)


The basic properties of the polynomials p n (x) that satisfy the orthogonality relations
$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$
hold also for the polynomials that satisfy the orthogonality relations of a more general form, which can be expressed in terms of Stielties integrals
$$ \int_a^b {{p_n}(x)} {p_m}(x)dw(x) = 0\quad (m \ne n), $$
where w(x) is a monotonic nondecreasing function (usually called the distribution function). The orthogonality relation (2.0.2) is reduced to (2.0.1) in the case when the function w(x) has a derivative on (a, b) and w′(x) = ϱ(x). For solving many problems orthogonal polynomials are used that satisfy the orthogonality relations (2.0.2) in the case when w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i . In this case the orthogonality relation (2.0.2) can be rewritten in the form
$$ \sum\limits_i {{p_n}({x_i})pm} ({x_i}){\rho_i} = 0\quad (m \ne n). $$


Difference Equation Orthogonal Polynomial Recursion Relation Discrete Variable Hermite Polynomial 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Arnold F. Nikiforov
    • 1
  • Vasilii B. Uvarov
    • 1
  • Sergei K. Suslov
    • 2
  1. 1.M.V. Keldysh Institute of Applied MathematicsAcademy of Sciences of the USSRMoscowUSSR
  2. 2.Kurchatov Institute of Atomic EnergyMoscowUSSR

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