# Gauss series

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##### 1: 15.2 Definitions and Analytical Properties
###### §15.2(i) GaussSeries
The hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss seriesOn 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 $F\left(-m,b;-m;z\right)$ is equal to the first $m+1$ terms of the Maclaurin series for $(1-z)^{-b}$.
##### 2: 15.19 Methods of Computation
The Gauss series (15.2.1) converges for $|z|<1$. … 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 $\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)$ 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
For 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
For 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
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
15.16.1 $F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},$ $|z|<1$,
15.16.2 $(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},$ $|z|<1$.
15.16.4 $F\left({a,b\atop c};z\right)F\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}% {c^{2}(1-c^{2})}z^{2}F\left({1+a,1+b\atop 2+c};z\right)F\left({1-a,1-b\atop 2-% c};z\right)=1.$
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
• 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
Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form … is a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion … Another generalization of Ramanujan’s sum is the Gauss sum $G\left(n,\chi\right)$ associated with a Dirichlet character $\chi\pmod{k}$. …In particular, $G\left(n,\chi_{1}\right)=c_{k}\left(n\right)$. $G\left(n,\chi\right)$ is separable for some $n$ if …
##### 10: Bibliography M
• 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.