relation to Heun equation

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11: 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

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12: 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 $w_j={\mathrm{𝐻𝑝}}_{n_j,m_j}$, $j=1,2$, satisfy …and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

13: Bibliography B

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

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1st item.

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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2nd item, 1st item.

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

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§31.13.

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

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§31.12.

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14: Bibliography H

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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).

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Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.

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ISSN 0022-2488, Document, FullText, MathReview, ZentralBlatt

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§31.17(ii), §31.17(ii).

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Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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External Links:

MathReview (Henrik Pedersen), ZentralBlatt

Referenced by:

§32.9(iii).

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Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.

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FullText, MathReview (P. Ungar), ZentralBlatt

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§28.29(iii).

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Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:

Reprinted by Dover in 1975.

External Links:

MathReview (Z. Opial), ZentralBlatt

Referenced by:

§29.3(vi).

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15: Bibliography R

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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MathReview (T. W. Kao), ZentralBlatt

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§2.8(iii).

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Heun’s Differential Equations. The Clarendon Press Oxford University Press, New York.

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ISBN 0-19-859695-2, MathReview (Harold Exton), ZentralBlatt

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§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.

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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

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External Links:

FullText, MathReview (A. D. Brjuno), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

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16: Bibliography F

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Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv., 38(1992), no. 3, pp. 621–635 (1992)

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ISSN 0373-2436, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

Referenced by:

§31.13.

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.

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ISSN 0010-3616, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§32.10(iv), §32.14, §32.6(v).

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17: Errata

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31.3.10 $$z^{-\alpha }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\alpha \left(\beta -ϵ\right)-\frac{\alpha }{a}\left(\beta -\delta \right);\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;\frac{1}{z})$$

31.3.11 $$z^{-\beta }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\beta \left(\alpha -ϵ\right)-\frac{\beta }{a}\left(\alpha -\delta \right);\beta ,\beta -\gamma +1,\beta -\alpha +1,\delta ;\frac{1}{z})$$

In both equations, the second entry in the $H\mathrm{\ell }$ has been corrected with an extra minus sign.

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_k=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}{u}_{k-1}$ was added to Equation (9.7.2).

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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18: Bibliography D

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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.

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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.

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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External Links:

ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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19: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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20: Bibliography G

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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External Links:

Document

Referenced by:

§19.35(ii).

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