Bernstein–Szegő polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

4: 18.3 Definitions

§18.3 Definitions

… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …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). … ►

Bessel polynomials

►Bessel polynomials are often included among the classical OP’s. …

5: 18.31 Bernstein–Szegő Polynomials

§18.31 Bernstein–Szegő Polynomials

►Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell }$ and positive when $-1\le x\le 1$. The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. …

6: 27.18 Methods of Computation: Primes

… ►An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). … ►The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. …

7: 18.14 Inequalities

… ►Equations (18.14.3) and (18.14.3_5) are Bernstein-type inequalities. … … ►

Legendre

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Jacobi

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Laguerre

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8: Bibliography K

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Unimodal sequences show that Lambert $W$ is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.

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External Links:

ISSN 0706-1994, MathReview (Fana Tangara), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:

ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

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External Links:

ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt

Referenced by:

§18.38(iii), §18.38(iii).

Encodings:

BibTeX

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Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

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Notes:

Electronic only

External Links:

Document, FullText

Referenced by:

§18.28(i), §18.28(i).

Encodings:

BibTeX

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

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9: Bibliography C

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A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.

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External Links:

MathReview (A. L. Whiteman), ZentralBlatt

Referenced by:

§24.10(iv).

Encodings:

BibTeX

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Stability properties of disk polynomials. Numer. Algorithms.

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External Links:

Document

Referenced by:

(18.14.3_5).

Encodings:

BibTeX

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A Bernstein-type inequality for the Jacobi polynomial. Proc. Amer. Math. Soc. 121 (3), pp. 703–709.

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External Links:

ISSN 0002-9939, FullText, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

BibTeX

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Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)

Referenced by:

§18.32.

Encodings:

BibTeX

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The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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10: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).