generalized Bessel polynomials

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31: 3.8 Nonlinear Equations

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§3.8(iv) Zeros of Polynomials

►The polynomial …No explicit general formulas exist when $n\ge 5$. … ►For further information on the computation of zeros of polynomials see McNamee (2007). … ►

Example. Wilkinson’s Polynomial

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32: 18.1 Notation

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Classical OP’s

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Laguerre: $L_n^{(\alpha )}\left(x\right)$ and $L_n\left(x\right)=L_n^{(0)}\left(x\right)$. ($L_n^{(\alpha )}\left(x\right)$ with $\alpha \ne 0$ is also called Generalized Laguerre.)

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Hahn Class OP’s

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Bessel: $y_n(x;a)$.

… ►Nor do we consider the shifted Jacobi polynomials: …

33: Bibliography R

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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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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

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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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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Applied Bessel Functions. Dover Publications Inc., New York.

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Notes:

Reprint of original Blackie & Son Limited, London, 1946.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.73(iii).

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:

English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

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34: Bibliography V

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:

Associated computer program available online

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ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.60(iii), §10.60(iii).

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Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.

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ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§14.29.

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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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A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

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35: Bibliography P

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

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Document, Code, ZentralBlatt

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1st item.

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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Document

Referenced by:

§25.18(i).

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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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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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36: Bibliography M

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Catastrophe optics of spheroidal drops and generalized rainbows. J. Quantit. Spec. and Rad. Trans. 63, pp. 341–351.

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Document

Referenced by:

§36.14(ii).

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The generalized integro-exponential function. Math. Comp. 44 (170), pp. 443–458.

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ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§8.19(v), §8.19(xi).

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

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Tables of Generalized Exponential Integrals. NPL Mathematical Tables, Vol. III, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. Fox)

Referenced by:

§8.25(v).

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Exceptional orthogonal polynomials.

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Notes:

Lecture at ICMAT School on Orthogonal Polynomials in Approximation Theory and Mathematical Physics, available at https://www.icmat.es/RT/optrim/school/lectures/Milson/icmat17.pdf

Referenced by:

§18.36(vi), §18.38(iii).

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37: Bibliography Y

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

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38: Software Index

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	Open Source													With Book						Commercial
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4.48(iii) General Precision	✓					✓	✓	✓				✓	✓								✓	✓	a		REDUCE
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10 Bessel Functions
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16 Generalized Hypergeometric Functions & Meijer G-Function
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18 Orthogonal Polynomials
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24 Bernoulli and Euler Polynomials
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39: Bibliography

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A general mode theory for the elliptic disk microstrip antenna. IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.

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Document

Referenced by:

5th item.

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

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Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

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External Links:

ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt

Referenced by:

§10.74(v).

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:

MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

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40: Bibliography S

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General relativity and quantum mechanics: towards a generalization of the Lambert $W$ function: a generalization of the Lambert $W$ function. Appl. Algebra Engrg. Comm. Comput. 17 (1), pp. 41–47.

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External Links:

ISSN 0938-1279, Document, FullText, MathReview (E. Capelas de Oliveira), ZentralBlatt

Referenced by:

§4.13, §4.13.

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Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

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Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.

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External Links:

ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.

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On numerical Bessel transformation. Comput. Phys. Comm. 16 (3), pp. 383–387.

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Notes:

Single-precision Fortran, maximum accuracy 15D.

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ISSN 0010-4655, Document, Code

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8th item.

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Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

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Document

Referenced by:

§34.13.

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