generalized hypergeometric series

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31: 17.4 Basic Hypergeometric Functions

… ►The series (17.4.1) is said to be balanced or Saalschützian when it terminates,

r=s

z=q

, and … ►The series (17.4.1) is said to be k-balanced when

r=s

and … ►The series (17.4.1) is said to be well-poised when

r=s

and … ►The series (17.4.1) is said to be very-well-poised when

r=s

, (17.4.11) is satisfied, and … ►The series (17.4.1) is said to be nearly-poised when

r=s

and …

32: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►

§31.11(ii) General Form

… ►

§31.11(v) Doubly-Infinite Series

►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions.

33: Bibliography M

… ►

A. R. Miller (1997) A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §16.10.
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

…

34: Bibliography O

… ►

O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt
Referenced by:: §25.8.
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

ⓘ

External Links:: ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt
Referenced by:: §2.9(i).
Encodings:: BibTeX

… ►

J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii), §2.7(ii).
Encodings:: BibTeX

►

F. W. J. Olver (1994b) The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

ⓘ

External Links:: MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §2.11(iii), §8.19(viii).
Encodings:: BibTeX

…

35: Bibliography

… ►

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

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

ⓘ

External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

… ►

G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

ⓘ

External Links:: ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.12.
Encodings:: BibTeX

►

G. E. Andrews (1986)

q

-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, Vol. 66, Amer. Math. Soc., Providence, RI.

ⓘ

External Links:: ISBN 0-8218-0716-1, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.14, §17.14, §17.16, Ch.17, Profile George E. Andrews.
Encodings:: BibTeX

… ►

R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

ⓘ

External Links:: ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.
Encodings:: BibTeX

…

36: 19.15 Advantages of Symmetry

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

F_{D}

(Carlson (1961b)). … ►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)). ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. …

37: 3.10 Continued Fractions

… ►

§3.10(ii) Relations to Power Series

►Every convergent, asymptotic, or formal series … ►We say that it corresponds to the formal power series … ►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). … ►

Forward Series Recurrence Algorithm

…

38: 13.31 Approximations

… ►

§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. … ►

13.31.1

A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),

ⓘ

Defines:: $A_{n}(z)$ : 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

… ►

13.31.2

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

ⓘ

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

… ►

13.31.3

z^{a}U\left(a,1+a-b,z\right)=\lim_{n\to\infty}\frac{A_{n}(z)}{B_{n}(z)}.

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $n$ : nonnegative integer, $z$ : complex variable, $A_{n}(z)$ : expansion and $B_{n}(z)$ : expansion
Permalink:: http://dlmf.nist.gov/13.31.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

39: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

►The hypergeometric function

F\left(a,b;c;z\right)

is defined by the Gauss series …In general,

F\left(a,b;c;z\right)

does not exist when

c=0,-1,-2,\dots

. … ►On the circle of convergence,

|z|=1

, the Gauss series: … ►The same properties hold for

F\left(a,b;c;z\right)

, except that as a function of

c

F\left(a,b;c;z\right)

in general has poles at

c=0,-1,-2,\dots

. …

40: 16.8 Differential Equations

… ► … ►

§16.8(ii) The Generalized Hypergeometric Differential Equation

… ►We have the connection formula …(Note that the generalized hypergeometric functions on the right-hand side are polynomials in

z_{0}

.) … ►

§16.8(iii) Confluence of Singularities

…