# summation

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##### 1: 17.18 Methods of Computation
โบThe two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …
##### 2: 34.10 Zeros
โบSimilarly the $\mathit{6j}$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3j}$ symbols in the summation. …
##### 3: 34.13 Methods of Computation
โบMethods of computation for $\mathit{3j}$ and $\mathit{6j}$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …
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##### 6: 28.34 Methods of Computation
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Summation of the power series in §§28.6(i) and 28.15(i) when $\left|q\right|$ is small.

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Summation of the power series in §§28.6(ii) and 28.15(ii) when $\left|q\right|$ is small.

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Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of $q$ and $z$.

• ##### 7: 1.8 Fourier Series
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###### §1.8(iv) Poisson’s Summation Formula
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1.8.16 $\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{-(n+x)^{2}\omega}={\sqrt{\frac{\pi}{% \omega}}\*\left(1+2\sum_{n=1}^{\infty}{\mathrm{e}}^{-n^{2}{\pi}^{2}/\omega}% \cos\left(2n\pi x\right)\right)},$ $\Re\omega>0$.
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##### 9: 34.3 Basic Properties: $\mathit{3j}$ Symbol
โบIn the summations (34.3.16)–(34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1)–(34.2.3). Similar conventions apply to all subsequent summations in this chapter.
##### 10: Bibliography Z
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• Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
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• I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.