Jacobi%20polynomials

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1: 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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BibTeX

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Jacobi polynomials. II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, pp. 125–137.

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External Links:

Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.12, §18.17(ii), §18.18(vi).

Encodings:

BibTeX

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Jacobi polynomials. III. An analytic proof of the addition formula. SIAM. J. Math. Anal. 6, pp. 533–543.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.18(ii).

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

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2: 18.5 Explicit Representations

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Jacobi

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $P_n^{(\alpha ,\beta )}\left(x\right)$ when the conditions $\alpha >-1$ and $\beta >-1$ are not satisfied. …For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ we assume throughout this chapter that $\alpha >-1$ and $\beta >-1$, unless stated otherwise. …

3: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

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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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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

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4: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

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A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math. 37 (5), pp. 979–1007.

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External Links:

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

Referenced by:

§18.15(i), §18.16(ii).

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BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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BibTeX

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5: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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BibTeX

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.

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External Links:

ISSN 0036-1410, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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External Links:

ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

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6: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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A Bernstein-type inequality for the Jacobi polynomial. Proc. Amer. Math. Soc. 121 (3), pp. 703–709.

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External Links:

ISSN 0002-9939, FullText, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.14(i).

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BibTeX

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Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

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BibTeX

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

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

Referenced by:

§13.29(v), §15.19(v).

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BibTeX

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

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7: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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FullText

Referenced by:

§8.24(i).

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

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External Links:

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

Referenced by:

§18.27(v).

Encodings:

BibTeX

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8: 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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External Links:

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

Referenced by:

§18.18(v).

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BibTeX

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The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:

MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

§18.30(viii).

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BibTeX

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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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External Links:

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

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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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 (3), pp. 56–64.

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ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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BibTeX

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Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-5159-4, MathReview, ZentralBlatt

Referenced by:

Ch.35, Profile Donald St. P. Richards.

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BibTeX

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9: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

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New inequalities for the zeros of Jacobi polynomials. SIAM J. Math. Anal. 18 (6), pp. 1549–1562.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(vi).

Encodings:

BibTeX

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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.

External Links:

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

Referenced by:

2nd item, §3.5(v), §3.5(v).

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BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

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

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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10: Errata

… ►We have also incorporated material on continuous $q$-Jacobi polynomials, and several new limit transitions. … ►

The generalized hypergeometric function of matrix argument ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$, was linked inadvertently as its single variable counterpart ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$. Furthermore, the Jacobi function of matrix argument $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and the Laguerre function of matrix argument $L_{\nu }^{(\gamma )}\left(𝐓\right)$, were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$, and $L_{\nu }^{(\gamma )}\left(𝐓\right)$. In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q)$$

Originally the first argument to the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder.

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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