Szeg%C5%91%E2%80%93Askey%20polynomials

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11: 24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(ii) Euler Numbers and Polynomials

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${(-1)}^n{E}_{2n}>0$ .

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Table 24.2.2: Bernoulli and Euler polynomials.

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$n$	$B_n\left(x\right)$	$E_n\left(x\right)$
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12: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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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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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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The umbral calculus. Pure and Applied Mathematics, Vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York.

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ISBN 0-12-594380-6, MathReview (R. Askey)

Referenced by:

§18.2(xii).

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13: Bibliography P

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Zonal Polynomials of Order $1$ Through $12$ . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

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MathReview

Referenced by:

§35.11.

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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6th item.

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Primzahlverteilung. Die Grundlehren der mathematischen Wissenschaften, Vol. 91, Springer-Verlag, Berlin-Göttingen-Heidelberg (German).

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Notes:

Reprinted in 1978.

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MathReview (A. E. Ingham), ZentralBlatt

Referenced by:

§27.11.

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Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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Document, ZentralBlatt

Referenced by:

§8.24(ii).

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14: Bibliography L

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Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

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Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

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ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

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15: Bibliography K

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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

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Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§3.8(iii).

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The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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ISSN 0377-0427, Document, Link, MathReview Entry

Referenced by:

(18.9.18_5).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

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Notes:

Electronic only

External Links:

Document, FullText

Referenced by:

§18.28(i), §18.28(i).

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16: Bibliography M

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Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.

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Document, MathReview (D.-N. Verma), ZentralBlatt

Referenced by:

§20.12(i).

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On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§14.17(i).

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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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ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt

Referenced by:

§27.13(iv).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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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).

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17: Bibliography C

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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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ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

Referenced by:

§16.24(ii).

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Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:

Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).

External Links:

CAOP

Referenced by:

1st item.

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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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ISSN 0036-1410, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.38(iii).

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

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Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

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ISBN 3-540-62838-X, MathReview, ZentralBlatt

Referenced by:

§10.73(i).

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18: 3.9 Acceleration of Convergence

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Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
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4	0.82221 76684 88	0.82246 28314 41	0.82246 69467 93	0.82246 70314 36	0.82246 70333 75
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9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
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19: Bibliography V

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Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

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ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt

Referenced by:

§8.10.

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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20: Bibliography S

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Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model. In Proceedings of the Conference on Mathematical Analysis and its Applications (Inst. Engrs., Mysore, 1977), Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.

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External Links:

MathReview

Referenced by:

§12.17.

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

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

Referenced by:

§28.8(iv).

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Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

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External Links:

FullText, MathReview (M. Strutt), ZentralBlatt

Referenced by:

§28.8(ii).

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

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A computer implementation of the Askey-Wilson scheme. Technical Report 13 Vrije Universteit Amsterdam.

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FullText

Referenced by:

CAOP (website).

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