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1: 12.16 Mathematical Applications
Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. …
2: 18.3 Definitions
  • 2.

    With the property that { p n + 1 ( x ) } n = 0 is again a system of OP’s. See §18.9(iii).

  • Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
    Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
    In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials T n ( x ) , n = 0 , 1 , , N , are orthogonal on the discrete point set comprising the zeros x N + 1 , n , n = 1 , 2 , , N + 1 , of T N + 1 ( x ) : … For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). …
    3: 18.19 Hahn Class: Definitions
    Hahn, Krawtchouk, Meixner, and Charlier
    Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.
    p n ( x ) X w x h n
    A special case of (18.19.8) is w ( 1 / 2 ) ( x ; π / 2 ) = π cosh ( π x ) .
    4: 3.11 Approximation Techniques
    They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. When n > 0 and 0 j n , 0 k n , … The c n in (3.11.11) can be calculated from (3.11.10), but in general it is more efficient to make use of the orthogonal property (3.11.9). …
    5: 14.17 Integrals
    §14.17(iii) Orthogonality Properties
    6: 18.35 Pollaczek Polynomials
    Then … For type 3 orthogonality (18.35.5) generalizes to …
    7: 18.25 Wilson Class: Definitions
    Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
    OP p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
    8: 18.34 Bessel Polynomials
    §18.34(ii) Orthogonality
    9: 28.31 Equations of Whittaker–Hill and Ince
    More important are the double orthogonality relations for p 1 p 2 or m 1 m 2 or both, given by …
    10: 18.5 Explicit Representations
    However, in these circumstances the orthogonality property (18.2.1) disappears. … …