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21: 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).

Encodings:

BibTeX

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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22: Bibliography N

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Numerical Algorithms Group, Ltd..

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Notes:

The NAG Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.

Referenced by:

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

Encodings:

BibTeX

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Notes:

Double precision. In Japanese, with some English.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Eigenstates, coherent states, and uncertainty products for the Morse oscillator. Phys. Rev. A (3) 19 (2), pp. 438–444.

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External Links:

ISSN 1050-2947, Document, Link, MathReview Entry

Referenced by:

(18.39.17), (18.39.18).

Encodings:

BibTeX

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Notes:

Single and double precision. See Kahaner et al. (1989).

External Links:

GAMS (package NMS)

Referenced by:

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

Encodings:

BibTeX

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Notes:

A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.

External Links:

Home Page

Referenced by:

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

Encodings:

BibTeX

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23: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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C Mathematical Function Handbook. McGraw-Hill, Inc., New York.

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Notes:

Includes diskette containing a large collection of mathematical function software written in C; Double-precision, maximum accuracy 9S.

External Links:

ISBN 00-791-1158-0

Referenced by:

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

Encodings:

BibTeX

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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External Links:

ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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24: Bibliography V

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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

Encodings:

BibTeX

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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

External Links:

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

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

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Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A 57 (1), pp. 190–197.

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External Links:

FullText

Referenced by:

§5.20.

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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25: Bibliography

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Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§24.13(i), §24.13(i).

Encodings:

BibTeX

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Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

(9.11.10), (9.11.5), (9.11.6), (9.11.7), (9.11.8), (9.11.9), §9.11(iv), §9.11(iv).

Encodings:

BibTeX

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Arb: A C Library for Arbitrary Precision Ball Arithmetic.

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Notes:

An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.

Referenced by:

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

Encodings:

BibTeX

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An integral of products of ultraspherical functions and a $q$-extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.

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External Links:

ISSN 0024-6107, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§10.22(iv), §14.30(ii).

Encodings:

BibTeX

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Center for Algorithms and Interactive Scientific Software.

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

Referenced by:

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

Encodings:

BibTeX

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26: Bibliography D

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Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

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

Encodings:

BibTeX

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.

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External Links:

MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

(12.12.4), §12.12, §18.17(iii), §18.17(iii).

Encodings:

BibTeX

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

BibTeX

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27: Bibliography K

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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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Tom Koornwinder’s Personal Collection of Maple Procedures

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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Algorithm 910: a portable C++ multiple-precision system for special-function calculations. ACM Trans. Math. Software 37 (4), pp. Art. 45, 27.

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Notes:

This system provides a uniform interface in C++, named e_float, to perform arithmetic operations and to calculate mathematical functions that are implemented in several different MP packages. It is a free, open-source, multiple precision package that allows the computation of many special functions. It supports calculations with 30….300 decimal digits and is interoperable with Microsoft’s CLR, Python and Mathematica.

External Links:

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

Referenced by:

Erratum (V1.0.9) for References, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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28: 19.16 Definitions

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19.16.7 $$\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\prod }_{j=1}^5\sqrt{t+x_j}}.$$

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Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/19.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

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19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

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Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

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