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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
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, 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, normalizations, and parameter constraints.
p n ( x ) X w x h n
k n = ( 2 sin ϕ ) n n ! .
4: 18.34 Bessel Polynomials
§18.34(ii) Orthogonality
5: 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). …
6: 14.17 Integrals
§14.17(iii) Orthogonality Properties
7: 18.35 Pollaczek Polynomials
§18.35(ii) Orthogonality
§18.35(iii) Other Properties
8: 18.27 q -Hahn Class
All these systems of OP’s have orthogonality properties of the form …Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … They are defined by their q -hypergeometric representations, followed by their orthogonality properties. …
9: 18.25 Wilson Class: Definitions
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
10: 18.5 Explicit Representations
However, in these circumstances the orthogonality property (18.2.1) disappears. … …