relations to confluent hypergeometric functions and generalized hypergeometric functions

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21: Bibliography B

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G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 1st item.
Encodings:: BibTeX

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G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 2nd item, 1st item.
Encodings:: BibTeX

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H. Buchholz (1969) The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Springer-Verlag, New York.

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Notes:: Translated from the German by H. Lichtbau and K. Wetzel
External Links:: MathReview, ZentralBlatt
Referenced by:: §12.17, §12.17, Ch.12, §13.1, §13.10(v), §13.14(i), §13.16(ii), §13.16(iii), §13.23(i), §13.24, §13.4(i), §13.4(ii), §13.4(iii), §13.6(iv), §13.9(i), §13.9(ii), Ch.13, §33.6, Notations M.
Encodings:: BibTeX

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W. Bühring (1988) An analytic continuation formula for the generalized hypergeometric function. SIAM J. Math. Anal. 19 (5), pp. 1249–1251.

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External Links:: ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

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W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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External Links:: ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt
Referenced by:: §16.4(ii), §16.8(ii).
Encodings:: BibTeX

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… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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Guide to Available Mathematical Software

A cross index of mathematical software in use at NIST.

23: Bibliography L

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S. Lewanowicz (1985) Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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S. Lewanowicz (1987) Corrigenda: “Recurrence relations for hypergeometric functions of unit argument” [Math. Comp. 45 (1985), no. 172, 521–535; MR 86m:33004]. Math. Comp. 48 (178), pp. 853.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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J. L. López and E. Pérez Sinusía (2014) New series expansions for the confluent hypergeometric function

M(a,b,z)

. Appl. Math. Comput. 235, pp. 26–31.

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External Links:: ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §13.11, §13.11.
Encodings:: BibTeX

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Y. L. Luke (1959) Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:: FullText, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §13.24(ii).
Encodings:: BibTeX

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Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:: MathReview (F. Gotze), ZentralBlatt
Referenced by:: §13.31(iii).
Encodings:: BibTeX

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24: 18.17 Integrals

… ►For the beta function

\mathrm{B}\left(a,b\right)

see §5.12, and for the confluent hypergeometric function

{{}_{1}F_{1}}

see (16.2.1) and Chapter 13. … ►For the confluent hypergeometric function

{{}_{1}F_{1}}

see (16.2.1) and Chapter 13. … ►This generalizes (18.17.34). For the hypergeometric function

{{}_{2}F_{1}}

see §§15.1 and 15.2(i). … ►For the generalized hypergeometric function

{{}_{2}F_{2}}

see (16.2.1). …

25: Errata

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Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

The Gegenbauer function $C^{(\lambda)}_{\alpha}\left(z\right)$ , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C^{(\lambda)}_{n}\left(z\right)$ . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind $U$ on the right-hand side of (7.18.10) was inserted.

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Paragraph Confluent Hypergeometric Functions (in §10.16)

Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

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Subsection 13.8(iii)

A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions $U\left(a,b,z\right)$ and ${\mathbf{M}}\left(a,b,z\right)$ as $a\to\infty$ in $|\operatorname{ph}a|\leq\pi-\delta$ and $b$ and $z$ fixed.

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Subsection 13.29(v)

A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

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26: 33.16 Connection Formulas

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§33.16(i) $F_{\ell}$ and $G_{\ell}$ in Terms of $f$ and $h$

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§33.16(ii) $f$ and $h$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

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§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$

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§33.16(iv) $s$ and $c$ in Terms of $F_{\ell}$ and $G_{\ell}$ when $\epsilon>0$

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§33.16(v) $s$ and $c$ in Terms of $W_{\kappa,\mu}\left(z\right)$ when $\epsilon<0$

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27: 13.14 Definitions and Basic Properties

… ►Standard solutions are: … ►In general

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

are many-valued functions of

z

with branch points at

z=0

and

z=\infty

. … ►Although

M_{\kappa,\mu}\left(z\right)

does not exist when

2\mu=-1,-2,-3,\dots

, many formulas containing

M_{\kappa,\mu}\left(z\right)

continue to apply in their limiting form. … ►Except when

z=0

, each branch of the functions

\ifrac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right)}

and

W_{\kappa,\mu}\left(z\right)

is entire in

\kappa

and

\mu

. Also, unless specified otherwise

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

are assumed to have their principal values. …

28: 3.10 Continued Fractions

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§3.10(ii) Relations to Power Series

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Stieltjes Fractions

… ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents

C_{n}=A_{n}/B_{n}

, and is related to the forward series recurrence algorithm. …

29: Bibliography G

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F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

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L. Gatteschi (1990) New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.9(i), §13.9(ii).
Encodings:: BibTeX

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W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:: MathReview (H. O. Pollak), ZentralBlatt
Referenced by:: §5.6(i), §6.8, §7.8, §8.10.
Encodings:: BibTeX

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W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.8(vi).
Encodings:: BibTeX

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W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:: ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.29(iii), §15.19(iii).
Encodings:: BibTeX

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30: Bibliography N

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M. Nardin, W. F. Perger, and A. Bhalla (1992a) Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:: Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.
External Links:: ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. Nardin, W. F. Perger, and A. Bhalla (1992b) Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:: Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.
External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §13.32(iii).
Encodings:: BibTeX

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National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §28.1, §28.26(i), 6th item, Notations S, Notations S.
Encodings:: BibTeX

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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

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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

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