Einstein summation convention for vectors

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21: 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. …

22: 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. …

23: 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). …

24: 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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25: 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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26: Bibliography M

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Vector Calculus. 4th edition, W. H. Freeman & Company, New York.

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Notes:

Fifth edition printed in 2003.

External Links:

ISBN 0-7167-2445-6, ZentralBlatt

Referenced by:

§1.5(i), §1.5(iii), §1.5(v), §1.5(vi), §1.6(i), §1.6(iii), §1.6(iv), §1.6(v).

Encodings:

BibTeX

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

Encodings:

BibTeX

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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27: 1.2 Elementary Algebra

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§1.2(v) Matrices, Vectors, Scalar Products, and Norms

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Row and Column Vectors

… ►and the corresponding transposed row vector of length $n$ is … ►Two vectors $𝐮$ and $𝐯$ are orthogonal if … ►

Vector Norms

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28: 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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29: 18.26 Wilson Class: Continued

… ►Here we use as convention for (16.2.1) with $b_q=-N$, $a_1=-n$, and $n=0,1,\mathrm{\dots },N$ that the summation on the right-hand side ends at $k=n$. …

30: Software Index

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	Open Source													With Book						Commercial
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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