relation to generalized hypergeometric functions

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31: 14.5 Special Values

§14.5 Special Values

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14.5.1

\mathsf{P}^{\mu}_{\nu}\left(0\right)=\frac{2^{\mu}\pi^{1/2}}{\Gamma\left(\frac% {1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\nu-\frac{% 1}{2}\mu\right)},

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.6.1 (modified)
Referenced by:: §10.59, §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.5(i), §14.5 and Ch.14

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§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$

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§14.5(vi) Addendum to §14.5(ii): $\mu=0$ , $\nu=2$

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32: 18.17 Integrals

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18.17.34_5

\int_{0}^{\infty}{\mathrm{e}}^{-xz}L^{(\alpha)}_{m}\left(x\right)L^{(\alpha)}_% {n}\left(x\right)\,{\mathrm{e}}^{-x}\,x^{\alpha}\,\mathrm{d}x=\frac{\Gamma% \left(\alpha+m+1\right)\Gamma\left(\alpha+n+1\right)}{\Gamma\left(\alpha+1% \right)\,m!\,n!}\frac{z^{m+n}}{(z+1)^{\alpha+m+n+1}}{{}_{2}F_{1}}\left({-m,-n% \atop\alpha+1};z^{-2}\right),

\Re z>-1

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $z$ : complex variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Source:: Erdélyi et al. (1954a, 4.11(35))
Proof sketch:: First assume $z>-1$ . Make change of integration variable $x\to\ifrac{x}{(z+1)}$ . Twice substitute (18.18.12) with $\lambda=(z+1)^{-1}$ . Then use the orthogonality relation for the Laguerre polynomials, see Table 18.3.1. Finally perform analytic continuation with respect to $z$ .
Referenced by:: Erratum (V1.2.0) Section 18.17
Permalink:: http://dlmf.nist.gov/18.17.E34_5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

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33: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

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§19.25(iv) Theta Functions

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§19.25(vii) Hypergeometric Function

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34: Bibliography S

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L. J. Slater (1966) 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.
Encodings:: BibTeX

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D. Slepian (1964) Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J. 43, pp. 3009–3057.

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External Links:: ISSN 0005-8580, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

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F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when

p=q+1

. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:: Document, MathReview (C. S. Meijer), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

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C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.23(iv), Ch.14, §31.3(iii).
Encodings:: BibTeX

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K. Srinivasa Rao (1981) Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

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External Links:: Document
Referenced by:: §34.13.
Encodings:: BibTeX

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35: 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. …

36: 16.23 Mathematical Applications

§16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§16.23(ii) Random Graphs

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§16.23(iv) Combinatorics and Number Theory

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37: 13.31 Approximations

§13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of

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

and

U\left(a,b,x\right)

that include the intervals

0\leq x\leq\alpha

and

\alpha\leq x<\infty

, respectively, where

\alpha

is an arbitrary positive constant. … ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). … ►

13.31.2

B_{n}(z)={{}_{2}F_{2}}\left({-n,n+1\atop a+1,b+1};-z\right).

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Defines:: $B_{n}(z)$ : expansion (locally)
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

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38: 16.3 Derivatives and Contiguous Functions

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§16.3(i) Differentiation Formulas

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§16.3(ii) Contiguous Functions

►Two generalized hypergeometric functions

{{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

are (generalized) contiguous if they have the same pair of values of

p

and

q

, and corresponding parameters differ by integers. If

p\leq q+1

, then any

q+2

distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations: …

39: 33.23 Methods of Computation

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§33.23(i) Methods for the Confluent Hypergeometric Functions

►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►

§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer

\ell

, provided that the recurrence is carried out in a stable direction (§3.6). … ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …

40: Bibliography C

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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

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F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

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B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric

R

-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §19.25(v), §19.29(i), §22.14(ii).
Encodings:: BibTeX

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F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert

W

function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.

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External Links:: ISSN 1053-587X, Document, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §4.44, §4.45(iii).
Encodings:: BibTeX

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W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §5.24(ii), §5.24(iii).
Encodings:: BibTeX

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