# finite sum

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## 1—10 of 75 matching pages

##### 1: 34.6 Definition: $\mathit{9j}$ Symbol
βΊThe $\mathit{9j}$ symbol may be defined either in terms of $\mathit{3j}$ symbols or equivalently in terms of $\mathit{6j}$ symbols: βΊ
34.6.1 $\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},$
βΊThe $\mathit{9j}$ 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
βΊFor 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
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25.15.3 $L\left(s,\chi\right)=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}% \right),$
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25.15.6 $G(\chi)\equiv\sum_{r=1}^{k-1}\chi(r)e^{2\pi ir/k}.$
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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
βΊWhen both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sumβΊwhere ${{}_{3}F_{2}}$ is defined as in §16.2. βΊFor 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
βΊThe $\mathit{6j}$ symbol can be expressed as the finite sumβΊFor 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
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25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,$
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25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$
βΊwhen both $H\left(s,z\right)$ and $H\left(z,s\right)$ are finite. …
##### 7: 5.16 Sums
βΊFor 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).
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##### 9: 27.10 Periodic Number-Theoretic Functions
βΊEvery function periodic (mod $k$) can be expressed as a finite Fourier series of the form …An example is Ramanujan’s sum: … βΊis a periodic function of $n\pmod{k}$ and has the finite Fourier-series expansion … βΊ $G\left(n,\chi\right)$ is separable for some $n$ if … βΊThe finite Fourier expansion of a primitive Dirichlet character $\chi\pmod{k}$ has the form …
##### 10: Bibliography R
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• K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.