Jacobi%20triple%20product

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►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). … ►

Jacobi on Other Intervals

…

3: 20.4 Values at $z$ = 0

… ►

20.4.3

\theta_{2}\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)\left(1+q^{2n}\right)^{2},

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome
Referenced by:: (20.10.3)
Permalink:: http://dlmf.nist.gov/20.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4 and Ch.20

►

20.4.4

\theta_{3}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+q^{2n-1}\right)^{2},

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome
Referenced by:: (20.10.2)
Permalink:: http://dlmf.nist.gov/20.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4 and Ch.20

►

20.4.5

\theta_{4}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-q^{2n-1}\right)^{2}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome
Referenced by:: (20.10.1)
Permalink:: http://dlmf.nist.gov/20.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4 and Ch.20

►

Jacobi’s Identity

►

20.4.6

\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome
A&S Ref:: 16.28.6
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

…

4: 20.7 Identities

… ►

20.7.10

\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(iii), §20.7 and Ch.20

… ►

§20.7(iv) Reduction Formulas for Products

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

►

20.7.34

\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/20.7.E34
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. See Walker (2012).
See also:: Annotations for §20.7(ix), §20.7 and Ch.20

…

5: Bibliography R

… ►

M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

ⓘ

External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

… ►

J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: (9.11.16), (9.11.17), §9.11(iii), §9.11(v).
Encodings:: BibTeX

►

W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: §9.11(iii), §9.11(v).
Encodings:: BibTeX

… ►

H. Rosengren (1999) Another proof of the triple sum formula for Wigner

9j

-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt
Referenced by:: §34.6.
Encodings:: BibTeX

…

6: 22.3 Graphics

… ►Line graphs of the functions

\operatorname{sn}\left(x,k\right)

\operatorname{cn}\left(x,k\right)

\operatorname{dn}\left(x,k\right)

\operatorname{cd}\left(x,k\right)

\operatorname{sd}\left(x,k\right)

\operatorname{nd}\left(x,k\right)

\operatorname{dc}\left(x,k\right)

\operatorname{nc}\left(x,k\right)

\operatorname{sc}\left(x,k\right)

\operatorname{ns}\left(x,k\right)

\operatorname{ds}\left(x,k\right)

, and

\operatorname{cs}\left(x,k\right)

for representative values of real

x

and real

k

illustrating the near trigonometric (

k=0

), and near hyperbolic (

k=1

) limits. … ►

\operatorname{sn}\left(x,k\right)

\operatorname{cn}\left(x,k\right)

, and

\operatorname{dn}\left(x,k\right)

as functions of real arguments

x

and

k

. … ►

See accompanying text — Figure 22.3.13: $\operatorname{sn}\left(x,k\right)$ for $k=1-e^{-n}$ , $n=0$ to 20, $-5\pi\leq x\leq 5\pi$ . Magnify 3D Help

►

…

7: 20.11 Generalizations and Analogs

… ►This is Jacobi’s inversion problem of §20.9(ii). … ►Each provides an extension of Jacobi’s inversion problem. … ►For

m=1,2,3,4

n=1,2,3,4

, and

m\neq n

, define twelve combined theta functions

\varphi_{m,n}\left(z,q\right)

by … ►

20.11.9

\varphi_{m,n}\left(z,q\right)=\varphi_{m,1}\left(z,q\right)\varphi_{1,n}\left(% z,q\right)=\frac{1}{\varphi_{n,m}\left(z,q\right)}=\frac{\varphi_{m,1}\left(z,% q\right)}{\varphi_{n,1}\left(z,q\right)}=\frac{\varphi_{1,n}\left(z,q\right)}{% \varphi_{1,m}\left(z,q\right)}.

ⓘ

Symbols:: $\varphi_{\NVar{n},\NVar{m}}\left(\NVar{z},\NVar{q}\right)$ : combined theta function, $m$ : integer, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §20.11(v), §20.11 and Ch.20

…

8: 22.4 Periods, Poles, and Zeros

… ►For example, the poles of

\operatorname{sn}\left(z,k\right)

, abbreviated as

\operatorname{sn}

in the following tables, are at

z=2mK+(2n+1)iK^{\prime}

. … ►Then: (a) In any lattice unit cell

\operatorname{pq}\left(z,k\right)

has a simple zero at

z=\mbox{p}

and a simple pole at

z=\mbox{q}

. (b) The difference between p and the nearest q is a half-period of

\operatorname{pq}\left(z,k\right)

. This half-period will be plus or minus a member of the triple

{K,iK^{\prime},K+iK^{\prime}}

; the other two members of this triple are quarter periods of

\operatorname{pq}\left(z,k\right)

. … ►For example,

\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)

. …

9: 22.13 Derivatives and Differential Equations

… ►Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … ►

22.13.1

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{sn}}^{2}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{sn}}^{2}\left(z,k\right)\right),

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Referenced by:: §22.13(iii), §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

►

22.13.2

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.7

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{% 2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{% dc}}^{2}\left(z,k\right)-k^{2}\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.10

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{ns}\left(z,k\right)\right)^{% 2}=\left({\operatorname{ns}}^{2}\left(z,k\right)-k^{2}\right)\left({% \operatorname{ns}}^{2}\left(z,k\right)-1\right),

ⓘ

Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

…

10: 20.5 Infinite Products and Related Results

§20.5 Infinite Products and Related Results

►

§20.5(i) Single Products

… ►

Jacobi’s Triple Product

… ►

§20.5(iii) Double Products

… ►

Jacobi%20triple%20product

1—10 of 333 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

§22.16(ii) Jacobi’s Epsilon Function

Integral Representations

§22.16(iii) Jacobi’s Zeta Function

Properties

2: 18.3 Definitions

§18.3 Definitions

Jacobi on Other Intervals