Van Vleck polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

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)

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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. …

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

… ►

Mean-Value

…

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

… ►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: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

… ►The

V(z)

are called Van Vleck polynomials and the corresponding

S(z)

Stieltjes polynomials. … ►If

t_{k}

is a zero of the Van Vleck polynomial

V(z)

, corresponding to an

n

th degree Stieltjes polynomial

S(z)

, and

z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}

are the zeros of

S^{\prime}(z)

(the derivative of

S(z)

), then

t_{k}

is either a zero of

S^{\prime}(z)

or a solution of the equation … ►See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials. …

6: Bibliography V

… ►

A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.

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External Links:: Document, ZentralBlatt
Referenced by:: §30.16(ii), §30.18(iii).
Encodings:: BibTeX

… ►

A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

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Notes:: For software, see Van Buren (website)
External Links:: ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §28.36(iii).
Encodings:: BibTeX

►

Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).
Encodings:: BibTeX

►

H. C. van de Hulst (1957) Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.

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External Links:: MathReview (P. M. Morse)
Referenced by:: §9.16.
Encodings:: BibTeX

►

H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.

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Referenced by:: §6.17.
Encodings:: BibTeX

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7: Bibliography

… ►

M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

… ►

A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.

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External Links:: ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §31.15(ii).
Encodings:: BibTeX

… ►

W. A. Al-Salam (1990) Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:: Link, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §18.3.
Encodings:: BibTeX

… ►

M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:: ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt
Referenced by:: §31.15(ii).
Encodings:: BibTeX

… ►

R. Askey (1989) Continuous

q

-Hermite Polynomials when

q>1

. In

q

-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:: MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §18.28(vii).
Encodings:: BibTeX

…

… … ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … ►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials. … ►

18 Orthogonal Polynomials

…

9: 1.12 Continued Fractions

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Van Vleck’s Theorem

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10: 27.18 Methods of Computation: Primes

… ►A practical version is described in Bosma and van der Hulst (1990). ►The AKS (Agrawal–Kayal–Saxena) algorithm is the first deterministic, polynomial-time, primality test. …