# Gauss series

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##### 1: 15.2 Definitions and Analytical Properties
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###### §15.2(i) GaussSeries
โบThe hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss seriesโบOn the circle of convergence, $|z|=1$, the Gauss series: … โบIn that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F\left(-m,b;-m;z\right)$ is equal to the first $m+1$ terms of the Maclaurin series for $(1-z)^{-b}$.
##### 2: 15.19 Methods of Computation
โบThe Gauss series (15.2.1) converges for $|z|<1$. … โบLarge 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
โบExpansions of the form $\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)$ 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: $\mathit{6j}$ Symbol
โบFor alternative expressions for the $\mathit{6j}$ 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: $\mathit{3j}$ Symbol
โบFor alternative expressions for the $\mathit{3j}$ 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
โบIf $n=2$, 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 $F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},$ $|z|<1$,
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15.16.2 $(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},$ $|z|<1$.
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15.16.4 $F\left({a,b\atop c};z\right)F\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}% {c^{2}(1-c^{2})}z^{2}F\left({1+a,1+b\atop 2+c};z\right)F\left({1-a,1-b\atop 2-% c};z\right)=1.$
โบFor 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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• J. L. López and N. M. Temme (2013) New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.
• ##### 9: 27.10 Periodic Number-Theoretic Functions
โบEvery function periodic (mod $k$) can be expressed as a finite Fourier series of the form … โบis a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion … โบAnother generalization of Ramanujan’s sum is the Gauss sum $G\left(n,\chi\right)$ associated with a Dirichlet character $\chi\pmod{k}$. …In particular, $G\left(n,\chi_{1}\right)=c_{k}\left(n\right)$. โบ $G\left(n,\chi\right)$ is separable for some $n$ if …
##### 10: Bibliography M
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• 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.