sums of products

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1: 27.1 Special Notation

$d, k, m, n$	positive integers (unless otherwise indicated).
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$\sum_{d\mathbin{\|}n}$ , $\prod_{d\mathbin{\|}n}$	sum, product taken over divisors of $n$ .
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$\sum_{p}$ , $\prod_{p}$	sum, product extended over all primes.
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2: 15.17 Mathematical Applications

… ►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

3: 10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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4: 10.67 Asymptotic Expansions for Large Argument

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

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5: 18.2 General Orthogonal Polynomials

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§18.2(v) Christoffel–Darboux Formula

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6: 34.4 Definition: $\mathit{6j}$ Symbol

… ►The

\mathit{6j}

symbol is defined by the following double sum of products of

\mathit{3j}

symbols: … ►

34.4.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},

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Symbols:: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: Erratum (V1.0.7) for Equation (34.4.2)
Permalink:: http://dlmf.nist.gov/34.4.E2
Encodings:: pMML, png, TeX
Errata (effective with 1.0.7):: Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.
Reported 2012-12-31 by Yu Lin
See also:: Annotations for §34.4 and Ch.34

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7: 26.15 Permutations: Matrix Notation

… ►The inversion number of

\sigma

is a sum of products of pairs of entries in the matrix representation of

\sigma

: … ►

26.15.11

\sum_{k=0}^{n}r_{n-k}(B){\left(x-k+1\right)_{k}}=\prod_{j=1}^{n}(x+b_{j}-j+1).

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $x$ : real variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $n$ : nonnegative integer, $B$ : subset and $r_{j}(B)$ : number of rook positions
Permalink:: http://dlmf.nist.gov/26.15.E11
Encodings:: pMML, png, TeX
Errata (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §26.15, §26.15 and Ch.26

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8: Bibliography D

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K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:: ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

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9: 1.11 Zeros of Polynomials

… ►The sum and product of the roots are respectively

-b/a

and

c/a

. …

10: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►For sums of products of

\mathit{3j}

symbols, see Varshalovich et al. (1988, pp. 259–262). …