DLMF: Untitled Document

§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. …

§35.4 Partitions and Zonal Polynomials

… ►

Normalization

… ►

Orthogonal Invariance

… ►

Summation

… ►

Mean-Value

…

… ►

A. Máté, P. Nevai, and W. Van Assche (1991) The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:: ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)
Referenced by:: §18.2(xi).
Encodings:: BibTeX

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A. M. Mathai (1993) A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.

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External Links:: ISBN 0-19-853595-3, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §16.18, §16.19, §16.20.
Encodings:: BibTeX

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L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

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S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

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G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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E. Bank and M. E. H. Ismail (1985) The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (W. A. Al-Salam)
Referenced by:: §18.39(iv), §18.39(iv).
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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I. Bloch, M. H. Hull, A. A. Broyles, W. G. Bouricius, B. E. Freeman, and G. Breit (1950) Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:: Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §33.7.
Encodings:: BibTeX

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§18.27(ii) $q$ -Hahn Polynomials

… ►

§18.27(iii) Big $q$ -Jacobi Polynomials

… ►

§18.27(iv) Little $q$ -Jacobi Polynomials

… ►

Little $q$ -Laguerre polynomials

… ►

§18.27(v) $q$ -Laguerre Polynomials

…

… ►

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). …

… ►

P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:: ZentralBlatt
Referenced by:: §7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).
Encodings:: BibTeX

…

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►

Table 27.2.1: Primes.

► ►►

$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…

► ►

Table 27.2.2: Functions related to division.

► ►►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

►

… ►where the

Q_{n}(z)

are monic polynomials (coefficient of highest power of

z

is

1

) satisfying … ►Next, let

p_{m}(z)

be the polynomials defined by

p_{m}(z)=0

for

m<0

, and … ►where

P_{m}(z)

and

Q_{m}(z)

are polynomials of degree

m

, with no common zeros. … ►where

P_{j,n-1}(z)

and

Q_{j,n}(z)

are polynomials of degrees

n-1

and

n

, respectively, with no common zeros. … ►where

\lambda

,

\mu

are constants, and

P_{n-1}(z)

,

Q_{n}(z)

are polynomials of degrees

n-1

and

n

, respectively, with no common zeros. …

… ►

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

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L. Chihara (1987) On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.

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External Links:: ISSN 0036-1410, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

►

T. S. Chihara (1978) An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.

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External Links:: ISBN 0-677-04150-0, MathReview (A. G. Law), ZentralBlatt
Referenced by:: §1.16(ix), (18.2.33), (18.2.36), (18.2.38), §18.2(v), §18.2(iii), §18.2(iv), §18.2(vii), §18.2(x), §18.20(i), (18.30.25), §18.30(ii), §18.30, §18.40(ii), Ch.18.
Encodings:: BibTeX

►

T. S. Chihara and M. E. H. Ismail (1993) Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:: ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

… ►

D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-62838-X, MathReview, ZentralBlatt
Referenced by:: §10.73(i).
Encodings:: BibTeX

…

Al-Salam%E2%80%93Chihara%20polynomials

11—20 of 329 matching pages

11: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

12: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

Normalization

Orthogonal Invariance

Summation

Mean-Value

13: Bibliography M

14: Bibliography B

15: 18.27 $q$ -Hahn Class

§18.27(ii) $q$ -Hahn Polynomials

§18.27(iii) Big $q$ -Jacobi Polynomials

§18.27(iv) Little $q$ -Jacobi Polynomials

Little $q$ -Laguerre polynomials

§18.27(v) $q$ -Laguerre Polynomials

16: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

17: Bibliography H

18: 27.2 Functions

19: 32.8 Rational Solutions

20: Bibliography C

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

Al-Salam%E2%80%93Chihara%20polynomials

11—20 of 329 matching pages

11: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

12: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

Normalization

Orthogonal Invariance

Summation

Mean-Value

13: Bibliography M

14: Bibliography B

15: 18.27 q -Hahn Class

§18.27(ii) q -Hahn Polynomials

§18.27(iii) Big q -Jacobi Polynomials

§18.27(iv) Little q -Jacobi Polynomials

Little q -Laguerre polynomials

§18.27(v) q -Laguerre Polynomials

16: 24.1 Special Notation

Bernoulli Numbers and Polynomials

Euler Numbers and Polynomials

17: Bibliography H

18: 27.2 Functions

19: 32.8 Rational Solutions

20: Bibliography C

15: 18.27 $q$ -Hahn Class

§18.27(ii) $q$ -Hahn Polynomials

§18.27(iii) Big $q$ -Jacobi Polynomials

§18.27(iv) Little $q$ -Jacobi Polynomials

Little $q$ -Laguerre polynomials

§18.27(v) $q$ -Laguerre Polynomials

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…