# Gauss sum

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##### 1: 27.10 Periodic Number-Theoretic Functions
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 … For a primitive character $\chi\pmod{k}$, $G\left(n,\chi\right)$ is separable for every $n$, and … Conversely, if $G\left(n,\chi\right)$ is separable for every $n$, then $\chi$ is primitive (mod $k$). …
##### 2: 20.11 Generalizations and Analogs
###### §20.11(i) GaussSum
For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by
20.11.2 $\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum\limits_{k=0}^{n-1}e^{-\pi ik^{% 2}m/n}=\frac{e^{-\pi i/4}}{\sqrt{m}}\sum\limits_{j=0}^{m-1}e^{\pi ij^{2}n/m}=% \frac{e^{-\pi i/4}}{\sqrt{m}}G(-n,m).$
##### 3: 5.16 Sums
For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
##### 7: 34.2 Definition: $\mathit{3j}$ Symbol
where ${{}_{3}F_{2}}$ is defined as in §16.2. 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).
##### 8: 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).
##### 9: 15.2 Definitions and Analytical Properties
15.2.1 $F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b% \right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}% {c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{\Gamma\left(a\right)\Gamma% \left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s% \right)}{\Gamma\left(c+s\right)s!}z^{s},$
15.2.4 $F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right% )_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}^{m}(-1)^{n}\genfrac{(}{)}{0.% 0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}z^{n}.$
##### 10: 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 $(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},$ $|z|<1$.
15.16.3 $F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),$ $|z|<1$, $|\zeta|<1$, $|z+\zeta-z\zeta|<1$.