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

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

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Defines:

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

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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

►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … ►For exact values of the coefficients of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L^{(n)}\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

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 $z_1^{\prime },z_2^{\prime },\mathrm{\dots },z_{n-1}^{\prime }$ 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

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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

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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

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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

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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

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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

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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

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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

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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

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), §18.28(ii), Ch.18, Profile Richard A. Askey.

Encodings:

BibTeX

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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

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8: Tom H. Koornwinder

… … ►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

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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. …