Gauss series

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1: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … ►On the circle of convergence, $|z|=1$, the Gauss series: … ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

2: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …

3: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

§16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

… ►Expansions of the form ${\sum }_{n=1}^{\mathrm{\infty }}{(\pm 1)}^n{}_p{}^{}F_{p+1}^{}(𝐚;𝐛;-n^2{z}^2)$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

4: 34.4 Definition: $\mathit{6}j$ Symbol

… ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

5: 34.2 Definition: $\mathit{3}j$ Symbol

… ►For alternative expressions for the $\mathit{3}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_3{}^{}F_2^{}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

6: 19.19 Taylor and Related Series

… ►If $n=2$, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). …

7: 15.16 Products

§15.16 Products

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15.16.1 $$F(\genfrac{}{}{0pt}{}{a,b}{c-\frac{1}{2}};z)F(\genfrac{}{}{0pt}{}{c-a,c-b}{c+\frac{1}{2}};z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(c\right)}_s}{{\left(c+\frac{1}{2}\right)}_s}{A}_s{z}^s,$$ ,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E1

Encodings:

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See also:

Annotations for §15.16 and Ch.15

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15.16.2 $${(1-z)}^{a+b-c}F(2a,2b;2c-1;z)=\sum _{s=0}^{\mathrm{\infty }}{A}_s{z}^s,$$ .

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $A_s$: coefficients

Permalink:

http://dlmf.nist.gov/15.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.16 and Ch.15

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15.16.4 $$F(\genfrac{}{}{0pt}{}{a,b}{c};z)F(\genfrac{}{}{0pt}{}{-a,-b}{-c};z)+\frac{ab(a-c)(b-c)}{c^2(1-c^2)}{z}^{2}F(\genfrac{}{}{0pt}{}{1+a,1+b}{2+c};z)F(\genfrac{}{}{0pt}{}{1-a,1-b}{2-c};z)=1.$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.16

Permalink:

http://dlmf.nist.gov/15.16.E4

Encodings:

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See also:

Annotations for §15.16 and Ch.15

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

8: Bibliography L

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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9: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form … ►is a periodic function of $n(modk)$ and has the finite Fourier-series expansion … ►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. …In particular, $G(n,{\chi }_1)=c_k\left(n\right)$. ► $G(n,\chi )$ is separable for some $n$ if …

10: Bibliography M

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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