which are also Sheffer polynomials

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1: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

… ►

Normalization

… ►

Orthogonal Invariance

… ►

Summation

… ►

Mean-Value

…

2: 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

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

ⓘ

Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : 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. …

3: 24.1 Special Notation

… ►

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

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►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). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). … ►

Bessel polynomials

…

5: 18.2 General Orthogonal Polynomials

… ►

Sheffer Polynomials

… ► … ►In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system) ►The Bernoulli polynomials

B_{n}\left(x\right)

and Euler polynomials

E_{n}\left(x\right)

are examples of Sheffer polynomials which are not OP’s, see the generating functions (24.2.3) and (24.2.8). For other examples of Sheffer polynomials, not in DLMF, see Roman (1984). …

… ►Also included is a website (CAOP) operated by a university department. Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. … ►

•

CAOP (website). Computer Algebra and Orthogonal Polynomials.

…

7: 18.41 Tables

… ►

§18.41(i) Polynomials

►For

P_{n}\left(x\right)

(

=\mathsf{P}_{n}\left(x\right)

) see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates

T_{n}\left(x\right)

U_{n}\left(x\right)

L_{n}\left(x\right)

, and

H_{n}\left(x\right)

for

n=0(1)12

. …The precision is 10D, except for

H_{n}\left(x\right)

which is 6-11S. … ►See also Abramowitz and Stegun (1964, Tables 25.4, 25.9, and 25.10). …

8: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. … Wilson), introduced the Askey-Wilson polynomials. Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. Another significant contribution was the Askey-Gasper inequality for Jacobi polynomials which was published in Positive Jacobi polynomial sums. II (with G. … ►Additional books for which Askey served as author or editor include Orthogonal Polynomials and Special Functions, published by SIAM in 1975, Theory and application of special functions, published by Academic Press in 1975, Special Functions: Group Theoretical Aspects and Applications (with T. …

9: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). …

10: 6.17 Physical Applications

… ►For applications in astrophysics, see also van de Hulst (1980). Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.