Rogers–Fine identity

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31—40 of 153 matching pages

31: Bibliography M

… ►

S. C. Milne (1985b) An elementary proof of the Macdonald identities for

A_{l}^{(1)}

. Adv. in Math. 57 (1), pp. 34–70.

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External Links:: ISSN 0001-8708, Document, MathReview (Y. Lehrer-Ilamed), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

L. Moser and M. Wyman (1958a) Asymptotic development of the Stirling numbers of the first kind. J. London Math. Soc. 33, pp. 133–146.

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External Links:: Document, MathReview (N. J. Fine), ZentralBlatt
Referenced by:: §26.1, §26.8(vii).
Encodings:: BibTeX

…

$x, y$	real variables.
…
$\mathbf{I}$	identity matrix
…

34: 20.11 Generalizations and Analogs

… ►This is the discrete analog of the Poisson identity (§1.8(iv)). … ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►Similar identities can be constructed for

{{}_{2}F_{1}}\left(\tfrac{1}{3},\tfrac{2}{3};1;k^{2}\right)

{{}_{2}F_{1}}\left(\tfrac{1}{4},\tfrac{3}{4};1;k^{2}\right)

, and

{{}_{2}F_{1}}\left(\tfrac{1}{6},\tfrac{5}{6};1;k^{2}\right)

. …

35: 21.7 Riemann Surfaces

… ►

§21.7(ii) Fay’s Trisecant Identity

… ►where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). … ►

§21.7(iii) Frobenius’ Identity

…

36: 16.23 Mathematical Applications

… ►Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. In Petkovšek et al. (1996) tools are given for automated proofs of these identities.

37: 21.6 Products

… ►

§21.6(i) Riemann Identity

… ►Then …This is the Riemann identity. On using theta functions with characteristics, it becomes …Many identities involving products of theta functions can be established using these formulas. …

38: 25.10 Zeros

… ►

25.10.1

Z(t)\equiv\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\equiv$ : equals by definition, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $Z(t)$ : zeros function and $\vartheta(t)$ : function
Keywords:: definition
Source:: Titchmarsh (1986b, (4.17.2), p. 89)
Permalink:: http://dlmf.nist.gov/25.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

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25.10.2

\vartheta(t)\equiv\operatorname{ph}\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}it% \right)-\tfrac{1}{2}t\ln\pi

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\equiv$ : equals by definition, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\vartheta(t)$ : function and $\chi(s)$ : function
Keywords:: definition
Proof sketch:: Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta(t)\equiv-\frac{1}{2}\operatorname{ph}\chi(\frac{1}{2}+\mathrm{i}t)$ , by using (25.9.2), (1.9.18), (1.9.20), $\Gamma\left(\overline{z}\right)=\overline{\Gamma\left(z\right)}$ , and (4.8.10).
Permalink:: http://dlmf.nist.gov/25.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

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39: 27.8 Dirichlet Characters

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27.8.6

\sum_{r=1}^{\phi\left(k\right)}\chi_{r}\left(m\right)\overline{\chi}_{r}(n)=% \begin{cases}\phi\left(k\right),&m\equiv n\pmod{k},\\ 0,&\mbox{otherwise}.\end{cases}

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\overline{\NVar{z}}$ : complex conjugate, $\equiv$ : modular equivalence, $k$ : positive integer, $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.8 and Ch.27

►A Dirichlet character

\chi\pmod{k}

is called primitive (mod

k

) if for every proper divisor

d

k

(that is, a divisor

d<k

), there exists an integer

a\equiv 1\pmod{d}

, with

\left(a,k\right)=1

and

\chi\left(a\right)\neq 1

. … ►

27.8.7

\chi\left(a\right)=1\text{ for all $a\equiv 1$ (mod $d$)},

\left(a,k\right)=1

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Symbols:: $\chi\left(\NVar{n}\right)$ : Dirichlet character, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $\equiv$ : modular equivalence, $d$ : positive integer and $k$ : positive integer
Permalink:: http://dlmf.nist.gov/27.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §27.8 and Ch.27

…

40: 27.13 Functions

… ►

27.13.5

(\vartheta\left(x\right))^{2}=1+\sum_{n=1}^{\infty}r_{2}\left(n\right)x^{n}.

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Symbols:: $\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)$ : alternative notation, $r_{\NVar{k}}\left(\NVar{n}\right)$ : number of squares, $n$ : positive integer and $x$ : real number
Referenced by:: §27.13(iv)
Permalink:: http://dlmf.nist.gov/27.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §27.13(iv), §27.13 and Ch.27

►One of Jacobi’s identities implies that … ►Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. For more than 8 squares, Milne’s identities are not the same as those obtained earlier by Mordell and others.