# orthogonal functions with respect to weighted summation

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##### 1: 3.11 Approximation Techniques
Now suppose that $X_{k\ell}=0$ when $k\neq\ell$, that is, the functions $\phi_{k}(x)$ are orthogonal with respect to weighted summation on the discrete set $x_{1},x_{2},\dots,x_{J}$. … …
##### 2: 18.3 Definitions
###### §18.3 Definitions
For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … 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 $-1-\beta>\alpha>-1$ a finite system of Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$ is orthogonal on $(1,\infty)$ with weight function $w(x)=(x-1)^{\alpha}(x+1)^{\beta}$. … However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …
##### 3: 18.27 $q$-Hahn Class
###### §18.27 $q$-Hahn Class
Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. …
##### 4: Bibliography G
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.
• W. Gautschi (1996) Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.
• W. Gautschi (2009) Variable-precision recurrence coefficients for nonstandard orthogonal polynomials. Numer. Algorithms 52 (3), pp. 409–418.
• R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in ${\rm U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
• ##### 5: 18.20 Hahn Class: Explicit Representations
###### §18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions
For the definition of hypergeometric and generalized hypergeometric functions see §16.2. Here we use as convention for (16.2.1) with $b_{q}=-N$, $a_{1}=-n$, and $n=0,1,\ldots,N$ that the summation on the right-hand side ends at $k=n$. …(For symmetry properties of $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …