# finite sum

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##### 1: 34.6 Definition: $\mathit{9j}$ Symbol
The $\mathit{9j}$ symbol may be defined either in terms of $\mathit{3j}$ symbols or equivalently in terms of $\mathit{6j}$ symbols:
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
The $\mathit{9j}$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
##### 2: 15.15 Sums
For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …
##### 3: 25.15 Dirichlet $L$-functions
25.15.3 $L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}% \right),$
25.15.6 $G(\chi)\equiv\sum_{r=1}^{k-1}\chi(r)e^{2\pi ir/k}.$
25.15.10 $L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k-1}r% \chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$
##### 4: 34.2 Definition: $\mathit{3j}$ Symbol
When both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sumwhere ${{}_{3}F_{2}}$ 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).
##### 5: 34.4 Definition: $\mathit{6j}$ Symbol
###### §34.4 Definition: $\mathit{6j}$ Symbol
The $\mathit{6j}$ symbol can be expressed as the finite sumFor 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).
##### 6: 25.16 Mathematical Applications
25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\mathrm{d}x,$
25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\mathrm{d}x.$
when both $H\left(s,z\right)$ and $H\left(z,s\right)$ are finite. …
##### 7: 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).
##### 9: Bibliography R
• K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
• ##### 10: 27.10 Periodic Number-Theoretic Functions
Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form …An example is Ramanujan’s sum: … is a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion … $G\left(n,\chi\right)$ is separable for some $n$ if … The finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …