polynomial cases

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21: Bibliography N

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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External Links:

ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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22: 18.5 Explicit Representations

… ►For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ we assume throughout this chapter that $\alpha >-1$ and $\beta >-1$, unless stated otherwise. Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. …

23: 18.39 Applications in the Physical Sciences

… ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). …

24: 18.30 Associated OP’s

… ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). … ►In the monic case, the monic associated polynomials ${\widehat{p}}_n(x;c)$ of order $c$ with respect to the ${\widehat{p}}_n(x)$ are obtained by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to …

25: 18.35 Pollaczek Polynomials

… ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). …

26: 3.5 Quadrature

… ►In the case of Chebyshev weight functions $w(x)={(1-x)}^{\alpha }{(1+x)}^{\beta }$ on $[-1,1]$, with $\left|\alpha \right|=\left|\beta \right|=\frac{1}{2}$, the nodes $x_k$, weights $w_k$, and error constant ${\gamma }_n$, are as follows: … ►In case of the Jacobi polynomials we have $p_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/k_n$, $q_n(x)=P_n^{(\alpha ,\beta )}\left(x\right)/\sqrt{h_n}$, and …

27: 13.6 Relations to Other Functions

… ►Special cases are the error functions … ►and in the case that $b-2a$ is an integer we have …Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively … ►Special cases of §13.6(iv) are as follows. … ►

Laguerre Polynomials

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28: 18.7 Interrelations and Limit Relations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). … ►

18.7.25

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Source:

Andrews et al. (1999, (6.4.13))

Referenced by:

§18.7(i), §18.7(iii), Erratum (V1.2.0) for Equation (18.7.25)

Permalink:

http://dlmf.nist.gov/18.7.E25

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

We included the case $n=0$.

See also:

Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

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29: 18.40 Methods of Computation

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … … ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►This is a challenging case as the desired $w^{\mathrm{RCP}}(x)$ on $[-1,1]$ has an essential singularity at $x=-1$. …

30: 18.27 $q$-Hahn Class

… ►The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. …