solutions in terms of classical orthogonal polynomials

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1: 18.39 Physical Applications

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§18.39(i) Quantum Mechanics

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2: 18.40 Methods of Computation

… ►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. ►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

3: 18.38 Mathematical Applications

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Quadrature

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Differential Equations

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Integrable Systems

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Riemann–Hilbert Problems

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Radon Transform

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4: Bibliography M

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Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-82472-9, MathReview (Eric M. Opdam), ZentralBlatt

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§18.37(iii).

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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57 (1-2), pp. 215–237.

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ISSN 0377-0427, Document, MathReview (Leonid B. Golinskiĭ), ZentralBlatt

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§32.15.

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Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. In Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), Vol. 49, pp. 169–178.

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External Links:

ISSN 0377-0427, Document, Link, MathReview Entry

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Erratum (V1.0.28) for Table 18.3.1.

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Classical transcendental solutions of the Painlevé equations and their degeneration. Tohoku Math. J. (2) 56 (4), pp. 467–490.

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ISSN 0040-8735, Document, MathReview (Oleg Gleizer), ZentralBlatt

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§32.10(v), §32.10(vi).

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Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, pp. 37–65.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

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§32.8(iii).

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5: Bibliography L

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Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.

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Notes:

For a Maple implementation of the algorithm given in this paper for Jack and zonal polynomials see Zeilberger (website).

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ISSN 0010-3616, Link, MathReview (Kazuhiro Hikami), ZentralBlatt

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§35.12.

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Orthogonal polynomials, duality and association schemes. SIAM J. Math. Anal. 13 (4), pp. 656–663.

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ISSN 0036-1410, Document, MathReview (C. L. Parihar), ZentralBlatt

Referenced by:

§18.38(iii).

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Orthogonal Polynomials for Exponential Weights. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4, Springer-Verlag, New York.

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ISBN 0-387-98941-2, MathReview (Walter Van Assche), ZentralBlatt

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§18.32.

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

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§18.15(vi), §18.16(vi).

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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

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§24.11, §24.11.

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6: Bibliography B

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(iii).

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

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On the Associated Legendre Polynomials. In Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), pp. 43–50.

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MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.30.

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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt

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§18.32.

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

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§3.11(iv).

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7: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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FullText, MathReview (S. C. van Veen), ZentralBlatt

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§3.5(viii).

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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ISBN 0-8218-3446-0, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§18.33(i).

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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt

Referenced by:

§18.33(i).

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Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.

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Notes:

American Mathematical Society Colloquium Publications, Vol. 23

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MathReview (J. Burlak)

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Ch.14, §9.1, Notations A.

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8: Bibliography K

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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).

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Hypergeometric Orthogonal Polynomials and Their $q$-Analogues. Springer Monographs in Mathematics, Springer-Verlag, Berlin.

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Notes:

With a foreword by Tom H. Koornwinder. Based in part on Koekoek and Swarttouw (1998).

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ISBN 978-3-642-05013-8, Document, FullText, MathReview (Jeremy Lovejoy), ZentralBlatt

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(17.4.2), §18.1(iii), §18.22(ii), §18.26(iii), §18.27(i), §18.27(ii), §18.27(iii), §18.27(iv), §18.27(v), §18.28(i), §18.28(ii), §18.28(iii), §18.28(viii), Ch.18, Profile René F. Swarttouw, Profile Roelof Koekoek.

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Orthogonal polynomials and computer algebra. In Recent developments in complex analysis and computer algebra (Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.), Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.

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External Links:

FullText, MathReview Entry, ZentralBlatt

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CAOP (website).

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Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:

MathReview (R. A. Askey), ZentralBlatt

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§18.37(ii).

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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).

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9: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

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Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

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2nd item, §3.5(v), §3.5(v).

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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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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

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ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan)

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

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Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.

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External Links:

MathReview (Sven Ehrich), ZentralBlatt

Referenced by:

§3.5(v).

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Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.

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Notes:

OPQ, a package of Matlab routines for generating classical and Sobolev orthogonal polynomials, accompanies this book.

External Links:

OPQ, ISBN 0-19-850672-4, MathReview Entry, ZentralBlatt

Referenced by:

§18.40, 3rd item, §3.5(v), Erratum (V1.0.28) for Table 18.3.1.

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