Vandermonde sum

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1: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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First $q$-Chu–Vandermonde Sum

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Second $q$-Chu–Vandermonde Sum

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Nonterminating Form of the $q$-Vandermonde Sum

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2: 15.4 Special Cases

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Chu–Vandermonde Identity

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Dougall’s Bilateral Sum

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15.4.25 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+n\right)\mathrm{\Gamma }\left(b+n\right)}{\mathrm{\Gamma }\left(c+n\right)\mathrm{\Gamma }\left(d+n\right)}=\frac{{\pi }^2}{\sin \left(\pi a\right)\sin \left(\pi b\right)}\frac{\mathrm{\Gamma }\left(c+d-a-b-1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(d-a\right)\mathrm{\Gamma }\left(c-b\right)\mathrm{\Gamma }\left(d-b\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.4(ii)

Permalink:

http://dlmf.nist.gov/15.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(ii), §15.4(ii), §15.4 and Ch.15

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3: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.4 $$det[a_{jk}]=\sum _{\mathrm{\ell }=1}^n{a}_{j\mathrm{\ell }}{A}_{j\mathrm{\ell }}.$$

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Symbols:

$det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $A_{jk}$: cofactor

Permalink:

http://dlmf.nist.gov/1.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

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1.3.9 $$det{[a_{jk}]}^2\le \left(\sum _{k=1}^n{a}_{1k}^2\right)\left(\sum _{k=1}^n{a}_{2k}^2\right)\mathrm{\dots }\left(\sum _{k=1}^n{a}_{nk}^2\right).$$

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Symbols:

$det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►

Vandermonde Determinant or Vandermondian

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1.3.19 $$\sum _{j,k=-\mathrm{\infty }}^{\mathrm{\infty }}\left|a_{j,k}-{\delta }_{j,k}\right|$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $j$: integer, $k$: integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.3.E19

Encodings:

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See also:

Annotations for §1.3(iii), §1.3 and Ch.1

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4: Bibliography D

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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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A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:

ZentralBlatt

Referenced by:

§15.4(ii).

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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