Heun%20operator

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11: 31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

►

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

denotes the solution of (31.2.1) that corresponds to the exponent

0

z=0

and assumes the value

1

there. … ►

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

… ►

§31.3(iii) Equivalent Expressions

… ►For example,

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

is equal to …

12: 31.18 Methods of Computation

§31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

13: Bibliography K

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E. G. Kalnins and W. Miller (1991a) Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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External Links:: ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt
Referenced by:: §31.16(ii).
Encodings:: BibTeX

►

E. G. Kalnins and W. Miller (1991b) Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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External Links:: ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt
Referenced by:: §31.16(ii).
Encodings:: BibTeX

►

E. G. Kalnins and W. Miller (1993) Orthogonal Polynomials on

n

-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §31.16(ii).
Encodings:: BibTeX

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A. Ya. Kazakov and S. Yu. Slavyanov (1996) Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.

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External Links:: ISSN 1073-2772, FullText, MathReview, ZentralBlatt
Referenced by:: §31.10(ii).
Encodings:: BibTeX

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T. H. Koornwinder (2006) Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:: FullText, MathReview Entry, ZentralBlatt
Referenced by:: §18.9(iii).
Encodings:: BibTeX

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14: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. … Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. …

15: 31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

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31.7.1

{{}_{2}F_{1}}\left(\alpha,\beta;\gamma;z\right)=\mathit{H\!\ell}\left(1,\alpha% \beta;\alpha,\beta,\gamma,\delta;z\right)=\mathit{H\!\ell}\left(0,0;\alpha,% \beta,\gamma,\alpha+\beta+1-\gamma;z\right)=\mathit{H\!\ell}\left(a,a\alpha% \beta;\alpha,\beta,\gamma,\alpha+\beta+1-\gamma;z\right).

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of

\mathit{H\!\ell}

to a

{{}_{2}F_{1}}

, with at least one free parameter, exist iff the pair

(a,p)

takes one of a finite number of values, where

q=\alpha\beta p

. … ►

31.7.2

\mathit{H\!\ell}\left(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1% ;z\right)={{}_{2}F_{1}}\left(\tfrac{1}{2}\alpha,\tfrac{1}{2}\beta;\gamma;1-(1-% z)^{2}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

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§31.7(ii) Relations to Lamé Functions

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16: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. ►

§31.16(ii) Heun Polynomial Products

… ►

17: Bibliography R

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M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:: ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

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A. Ronveaux (Ed.) (1995) Heun’s Differential Equations. The Clarendon Press Oxford University Press, New York.

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External Links:: ISBN 0-19-859695-2, MathReview (Harold Exton), ZentralBlatt
Referenced by:: §31.1, §31.12, §31.13, §31.2(v), §31.2(i), §31.2(iv), §31.3(ii), §31.4, §31.7(ii), §31.9(i), Ch.31, Profile Gerhard Wolf.
Encodings:: BibTeX

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G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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18: 31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

►Heun polynomials

w_{j}=\mathit{Hp}_{n_{j},m_{j}}

j=1,2

, satisfy …and the integration paths

\mathcal{L}_{1}

\mathcal{L}_{2}

are Pochhammer double-loop contours encircling distinct pairs of singularities

\{0,1\}

\{0,a\}

\{1,a\}

. …

19: Bibliography D

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A. Decarreau, M.-Cl. Dumont-Lepage, P. Maroni, A. Robert, and A. Ronveaux (1978a) Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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External Links:: MathReview (A. E. Danese), ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

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A. Decarreau, P. Maroni, and A. Robert (1978b) Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

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External Links:: ISSN 0037-959X, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

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B. Deconinck and J. N. Kutz (2006) Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.

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External Links:: ISSN 0021-9991, MathReview, ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

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N. Dunford and J. T. Schwartz (1988) Linear operators. Part II. Wiley Classics Library, John Wiley & Sons, Inc., New York.

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Notes:: Spectral theory. Selfadjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication
External Links:: ISBN 0-471-60847-5, MathReview Entry
Referenced by:: (1.18.68), §1.18(ix), §1.18(ix), §1.18(ix), §1.18(x).
Encodings:: BibTeX

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C. F. Dunkl (1989) Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.

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External Links:: ISSN 0002-9947, Link, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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20: Bibliography S

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S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:: ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

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B. D. Sleeman (1966a) Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:: Available from the library, University of Surrey, UK
External Links:: LibraryRecord
Referenced by:: §31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.
Encodings:: BibTeX

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B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:: Document, MathReview (J. D. DePree), ZentralBlatt
Referenced by:: §31.10(ii).
Encodings:: BibTeX

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A. O. Smirnov (2002) Elliptic Solitons and Heun’s Equation. In The Kowalevski Property (Leeds, UK, 2000), V. B. Kuznetsov (Ed.), CRM Proc. Lecture Notes, Vol. 32, pp. 287–306.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §31.8.
Encodings:: BibTeX

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H. Suzuki, E. Takasugi, and H. Umetsu (1998) Perturbations of Kerr-de Sitter black holes and Heun’s equations. Progr. Theoret. Phys. 100 (3), pp. 491–505.

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External Links:: ISSN 0033-068X, FullText, MathReview (Volker Perlick)
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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