expansions in series of hypergeometric functions

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31—40 of 63 matching pages

31: Bibliography T

… ►

N. M. Temme (1978) Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

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External Links:: ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.8(ii), §13.8(ii).
Encodings:: BibTeX

… ►

N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:: ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt
Referenced by:: §13.8(iii), §13.8(iii).
Encodings:: BibTeX

… ►

N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.
Encodings:: BibTeX

… ►

N.M. Temme and E.J.M. Veling (2022) Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

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External Links:: ISSN 0019-3577, Document, Link
Referenced by:: §13.8(iv).
Encodings:: BibTeX

… ►

N. M. Temme (2022) Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:: ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §13.8(iv).
Encodings:: BibTeX

…

32: 19.15 Advantages of Symmetry

… ►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s

F_{D}

(Carlson (1961b)). The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. … ►(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. … ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

33: 15.15 Sums

§15.15 Sums

►

15.15.1

\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.19(i)
Permalink:: http://dlmf.nist.gov/15.15.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §15.15 and Ch.15

►Here

z_{0}

(

{\neq 0}

) is an arbitrary complex constant and the expansion converges when

|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)

. … ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).

34: 8.27 Approximations

… ►

§8.27(i) Incomplete Gamma Functions

… ►

•

Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$ , and $\gamma\left(a,\omega z\right)$ with $0\leq\omega\leq 1$ ; see also Temme (1994b, §3).

►

•

Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$ -plane that exclude $z=0$ and are valid for $\left|\operatorname{ph}z\right|<\pi$ .

… ►

•

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$ .

… ►

•

Verbeeck (1970) gives polynomial and rational approximations for $E_{p}\left(x\right)=(e^{-x}/x)P(z)$ , approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$ .

35: Bibliography W

… ►

G. Wei and B. E. Eichinger (1993) Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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External Links:: ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

… ►

E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:: ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

… ►

F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

E. M. Wright (1940a) The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, pp. 389–408.

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External Links:: ISSN 0024-6115, Document, MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §16.11(ii).
Encodings:: BibTeX

…

36: Bibliography N

… ►

J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §13.28(i).
Encodings:: BibTeX

… ►

G. Nemes (2017a) Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:: ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.15.
Encodings:: BibTeX

►

G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

… ►

N. E. Nørlund (1955) Hypergeometric functions. Acta Math. 94, pp. 289–349.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §16.10, §16.8(ii), §16.8(ii).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1992) Diffraction Effects in Semiclassical Scattering. Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.

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Notes:: Also available electronically through Cambridge Books Online, ISBN 9780511599903.
External Links:: ISBN 9780521383189, Document
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

…

37: 2.11 Remainder Terms; Stokes Phenomenon

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►In this way we arrive at hyperasymptotic expansions. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

38: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►If

n=2

, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … ►Then

T_{N}

has at most one term if

N\leq 5

in the series for

R_{F}

. … ►

39: Bibliography C

… ►

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) 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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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

… ►

B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

… ►

T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

ⓘ

External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

… ►

J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function

J_{n}(z)

in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

… ►

R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert

W

function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.
Encodings:: BibTeX

…

40: 13.29 Methods of Computation

… ►

§13.29(i) Series Expansions

►Although the Maclaurin series expansion (13.2.2) converges for all finite values of

z

, it is cumbersome to use when

|z|

is large owing to slowness of convergence and cancellation. …However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). … ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of

M\left(n,b,x\right)

, when

b

and

x

are real and

n

is a positive integer. …