expansions in series of hypergeometric functions

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41: 3.10 Continued Fractions

… ►Every convergent, asymptotic, or formal series … ►For several special functions the

S

-fractions are known explicitly, but in any case the coefficients

a_{n}

can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … ►We say that it is associated with the formal power series

f(z)

in (3.10.7) if the expansion of its

n

th convergent

C_{n}

in ascending powers of

z

, agrees with (3.10.7) up to and including the term in

z^{2n-1}

n=1,2,3,\dots

. … ►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). … ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …

42: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

… ► … ►

Confluent Hypergeometric Functions

… ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors

\tfrac{1}{2}\pi<\left|\operatorname{ph}z\right|<\pi

one has to use the analytic continuation formula (13.2.12). … ►

§7.18(vi) Asymptotic Expansion

…

43: Bibliography H

… ►

P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

G. H. Hardy (1949) Divergent Series. Clarendon Press, Oxford.

ⓘ

Notes:: Reprinted by the American Mathematical Society in 2000.
External Links:: MathReview (R. P. Agnew), ZentralBlatt
Referenced by:: §1.15(ii), §1.15(viii).
Encodings:: BibTeX

… ►

E. Hille (1929) Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.9.
Encodings:: BibTeX

… ►

Y. P. Hsu (1993) Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

ⓘ

External Links:: ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §15.20(iii).
Encodings:: BibTeX

… ►

G. Hunter and M. Kuriyan (1976) Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.

ⓘ

External Links:: Document, MathReview (John P. Coleman)
Referenced by:: 5th item.
Encodings:: BibTeX

…

44: 2.10 Sums and Sequences

… ►Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … ►Hence … ►What is the asymptotic behavior of

f_{n}

n\to\infty

n\to-\infty

? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …

45: 14.15 Uniform Asymptotic Approximations

… ►In other words, the convergent hypergeometric series expansions of

\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)

are also generalized (and uniform) asymptotic expansions as

\mu\to\infty

, with scale

\ifrac{1}{\Gamma\left(j+1+\mu\right)}

j=0,1,2,\dots

; compare §2.1(v). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). … ►For convergent series expansions see Dunster (2004). … ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials

P_{n}\left(\cos\theta\right)

n\to\infty

with

\theta

fixed. …

46: Bibliography

… ►

G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen (1992b) Hypergeometric Functions and Elliptic Integrals. In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.), pp. 48–85.

ⓘ

External Links:: MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

… ►

G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

ⓘ

External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §17.6(ii).
Encodings:: BibTeX

… ►

K. Aomoto (1987) Special value of the hypergeometric function

{}_{3}F_{2}

and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

ⓘ

External Links:: ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

… ►

T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97127-0, MathReview, ZentralBlatt
Referenced by:: §17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.
Encodings:: BibTeX

… ►

R. Askey (1975b) Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.10(ii), §18.18(vii), Profile Richard A. Askey.
Encodings:: BibTeX

…

47: Bibliography S

… ►

M. J. Seaton (2002b) FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

ⓘ

Notes:: Single-precision Fortran code.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 10th item.
Encodings:: BibTeX

… ►

H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §12.13(i).
Encodings:: BibTeX

►

H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… ►

H. Skovgaard (1966) Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

ⓘ

External Links:: MathReview (A. K. Agarwal), ZentralBlatt
Referenced by:: §13.21(iv).
Encodings:: BibTeX

… ►

B. D. Sleeman (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

ⓘ

External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.17(i).
Encodings:: BibTeX

…

48: 15.16 Products

§15.16 Products

… ►where

A_{0}=1

and

A_{s}

s=1,2,\dots

, are defined by the generating function … ►

15.16.3

F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),

|z|<1

|\zeta|<1

|z+\zeta-z\zeta|<1

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

… ►

Generalized Legendre’s Relation

… ►For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).

49: Bibliography M

… ►

S. C. Milne (1985a) A

q

-analog of the

{}_{5}F_{4}(1)

summation theorem for hypergeometric series well-poised in

\mathit{SU}(n)

. Adv. in Math. 57 (1), pp. 14–33.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

►

S. C. Milne (1985d) A

q

-analog of hypergeometric series well-poised in

\mathit{SU}(n)

and invariant

G

-functions. Adv. in Math. 58 (1), pp. 1–60.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

►

S. C. Milne (1988) A

q

-analog of the Gauss summation theorem for hypergeometric series in

U(n)

. Adv. in Math. 72 (1), pp. 59–131.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

►

S. C. Milne (1994) A

q

-analog of a Whipple’s transformation for hypergeometric series in

U(n)

. Adv. Math. 108 (1), pp. 1–76.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

50: Bibliography B

… ►

W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

►

W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

ⓘ

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

… ►

W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

ⓘ

External Links:: MathReview
Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
Encodings:: BibTeX

… ►

J. L. Burchnall and T. W. Chaundy (1940) Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

ⓘ

External Links:: Document, MathReview (G. Szegő), ZentralBlatt
Referenced by:: §15.16, (16.15.4), §16.15, §16.16(i).
Encodings:: BibTeX

►

J. L. Burchnall and T. W. Chaundy (1941) Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

ⓘ

External Links:: Document, MathReview (G. Szegő), ZentralBlatt
Referenced by:: §16.16(i).
Encodings:: BibTeX

…