Chu–Vandermonde identity

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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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Rogers–Fine Identity

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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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3: Bibliography S

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On an identity of Ramanujan based on the hypergeometric series ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)$ . J. Number Theory 69 (2), pp. 125–134.

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External Links:

ISSN 0022-314X, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

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External Links:

ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

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External Links:

MathReview (H. Bateman), ZentralBlatt

Referenced by:

§28.1, 7th item, Notations S, Notations S.

Encodings:

BibTeX

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Spheroidal Wave Functions: Including Tables of Separation Constants and Coefficients. Technology Press of M. I. T. and John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0262190028, 0262692910, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

1st item, Ch.30.

Encodings:

BibTeX

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4: 1.3 Determinants

… ►If two rows (columns) of a determinant are identical, then the determinant is zero. … ►

Vandermonde Determinant or Vandermondian

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5: Preface

… ►Chu, A. …

6: 24.10 Arithmetic Properties

… ►where $m\equiv n\cancel{\equiv }0(modp-1)$. …valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer. … ►

24.10.8 $$N_{2n}\equiv 0(mod{p}^{\mathrm{\ell }}),$$

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Symbols:

$\equiv$: modular equivalence, $\mathrm{\ell }$: integer, $n$: integer and $p$: prime

Referenced by:

§24.10(iv)

Permalink:

http://dlmf.nist.gov/24.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.10(iv), §24.10 and Ch.24

►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. ►

24.10.9

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Symbols:

$E_n$: Euler numbers, $\equiv$: modular equivalence, $\mathrm{\ell }$: integer, $n$: integer and $p$: prime

Referenced by:

§24.10(iv)

Permalink:

http://dlmf.nist.gov/24.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.10(iv), §24.10 and Ch.24

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7: Bibliography D

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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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Novel identities for simple $n$-$j$ symbols. J. Mathematical Phys. 16, pp. 318–319.

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External Links:

Document, MathReview (M. J. Westwater)

Referenced by:

§34.5(vi).

Encodings:

BibTeX

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8: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. But $y^s\equiv x^{rs}\equiv x^{1+t\varphi \left(n\right)}\equiv x(modn)$, so $y^s$ is the same as $x$ modulo $n$. …

9: 36.9 Integral Identities

§36.9 Integral Identities

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36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integral identity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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10: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …