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1: 5.19 Mathematical Applications

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§5.19(i) Summation of Rational Functions

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2: 17.18 Methods of Computation

… ►The 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. …

3: 34.10 Zeros

… ►Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …

4: 34.13 Methods of Computation

… ►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ 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). …

5: 24.17 Mathematical Applications

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§24.17(i) Summation

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Euler–Maclaurin Summation Formula

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Boole Summation Formula

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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 $z$.

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7: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Ramanujan’s ${}_1{}^{}\psi _1^{}$ Summation

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Bailey’s Bilateral Summations

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8: 1.8 Fourier Series

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§1.8(iv) Poisson’s Summation Formula

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1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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9: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►In 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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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

BibTeX

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The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.

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External Links:

ISSN 0036-1410, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§4.41.

Encodings:

BibTeX

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