# orthogonal properties

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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
In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, of $T_{N+1}\left(x\right)$: … For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). …
##### 3: 18.19 Hahn Class: Definitions
###### Hahn, Krawtchouk, Meixner, and Charlier
$k_{n}=\frac{(2\sin\phi)^{n}}{n!}.$
##### 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 $0\leq j\leq n$, $0\leq k\leq n$, … 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). …
##### 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. …
##### 10: 18.5 Explicit Representations
However, in these circumstances the orthogonality property (18.2.1) disappears. … …