relations to other orthogonal polynomials

SearchAdvancedHelp

(0.018 seconds)

11—20 of 43 matching pages

11: 18.35 Pollaczek Polynomials

… ►

18.35.7 $${(1-z{\mathrm{e}}^{\mathrm{i}\theta })}^{-\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta })}^{-\lambda -\mathrm{i}{\tau }_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(\cos \theta ;a,b)z^n,$$ , .

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.8))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

…

12: 18.23 Hahn Class: Generating Functions

… ►

Hahn

…

13: 18.27 $q$-Hahn Class

§18.27 $q$-Hahn Class

… ►

18.27.26 $$\underset{q\to 1}{lim}\frac{{\tilde{h}}_n({(1-q^2)}^{\frac{1}{2}}x;q)}{{\left(1-q^2\right)}^{n/2}}=2^{-n}{H}_n\left(x\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, ${\tilde{h}}_n(x;q)$: discrete $q$-Hermite II polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.29.14))

Referenced by:

§18.27(vii), §18.27(vii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E26

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(vii), §18.27(vii), §18.27 and Ch.18

14: 3.5 Quadrature

… ► …

15: 18.28 Askey–Wilson Class

… ►

§18.28(ii) Askey–Wilson Polynomials

… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).

16: 18.30 Associated OP’s

… ►

§18.30(i) Associated Jacobi Polynomials

…

17: 18.34 Bessel Polynomials

… ►

§18.34(i) Definitions and Recurrence Relation

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►

§18.34(ii) Orthogonality

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : …

18: 18.36 Miscellaneous Polynomials

… ►

§18.36(ii) Sobolev Orthogonal Polynomials

… ►

§18.36(iii) Multiple Orthogonal Polynomials

►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►

§18.36(iv) Orthogonal Matrix Polynomials

… ►

§18.36(vi) Exceptional Orthogonal Polynomials

…

19: 18.3 Definitions

§18.3 Definitions

… ►

As given by a Rodrigues formula (18.5.5).

… ►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 $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: … ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

20: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

§18.7(iii) Limit Relations

►

Jacobi $\to$ Laguerre

… ►

Laguerre $\to$ Hermite

…