Jacobi–Abel addition theorem

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11—20 of 265 matching pages

12: Errata

… ►We have also incorporated material on continuous

q

-Jacobi polynomials, and several new limit transitions. … ►

Additions

Equation (16.16.5_5).

… ►

Equation (22.19.6)

22.19.6

x(t)=\operatorname{cn}\left(t\sqrt{1+2\eta},k\right)

Originally the term $\sqrt{1+2\eta}$ was given incorrectly as $\sqrt{1+\eta}$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k=1/\sqrt{2+\eta^{-1}}$ has been defined in the line above.

Reported 2014-05-02 by Svante Janson.

… ►

Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

… ►

Equation (22.6.7)

22.6.7

\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z,k% \right)-k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}% \left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{% \operatorname{dn}}^{4}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}% }^{4}\left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}

Originally the term $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}\left(z,k\right)$ was given incorrectly as $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,k\right)$ .

Reported 2014-02-28 by Svante Janson.

…

13: 18.10 Integral Representations

… ►

18.10.1

\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}% \left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{% (\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma% \left(\alpha+1\right)}{{\pi}^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right% )}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{% 2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\,\mathrm{d}\phi,

0<\theta<\pi

\alpha>-\tfrac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial and $n$ : nonnegative integer
A&S Ref:: 22.10.11
Referenced by:: §18.10(i), §18.10(i)
Permalink:: http://dlmf.nist.gov/18.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

… ►Generalizations of (18.10.1) for

P^{(\alpha,\beta)}_{n}

are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). … ►

Jacobi

… ►for the Jacobi, Laguerre, and Hermite polynomials. … ►

Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).

► ►►

$p_{n}(x)$	$g_{0}(x)$	$g_{1}(z,x)$	$g_{2}(z,x)$	$c$	Conditions
…

► …

14: 22.6 Elementary Identities

… ►

22.6.2

1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.9.4
Permalink:: http://dlmf.nist.gov/22.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.5

\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}}^{4}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.18.1
Permalink:: http://dlmf.nist.gov/22.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.8

\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}}^{2}\left(z,k% \right)-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{% nd}}^{2}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(% z,k\right)},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

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§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
…

► …

15: 18.30 Associated OP’s

… ►

§18.30(i) Associated Jacobi Polynomials

… ►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). … ►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real

x

-axis each multiplied by the polynomial product evaluated at the corresponding values of

x

, as in (18.2.3). … ►

Markov’s Theorem

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16: 18.1 Notation

… ►

Classical OP’s

►

Jacobi: $P^{(\alpha,\beta)}_{n}\left(x\right)$ .

… ►

Big $q$ -Jacobi: $P_{n}\left(x;a,b,c;q\right)$ .

►

Little $q$ -Jacobi: $p_{n}\left(x;a,b;q\right)$ .

… ►Nor do we consider the shifted Jacobi polynomials: …

17: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

18: 18.27 $q$ -Hahn Class

… ►

§18.27(iii) Big $q$ -Jacobi Polynomials

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From Big $q$ -Jacobi to Jacobi

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§18.27(iv) Little $q$ -Jacobi Polynomials

… ►

From Big $q$ -Jacobi to Little $q$ -Jacobi

… ►

From Little $q$ -Jacobi to Jacobi

…

19: 23.15 Definitions

… ►In §§23.15–23.19,

k

and

k^{\prime}

(\in\mathbb{C})

denote the Jacobi modulus and complementary modulus, respectively, and

q=e^{i\pi\tau}

(

\Im\tau>0

) denotes the nome; compare §§20.1 and 22.1. … ►

k=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},

►

k^{\prime}=\frac{{\theta_{4}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)}.

… ►If, in addition,

f(\tau)\to 0

q\to 0

, then

f(\tau)

is called a cusp form. … ►

23.15.7

J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},

ⓘ

Defines:: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

…

20: 22.13 Derivatives and Differential Equations

… ►

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

►

22.13.3

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

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : 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.E3
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

…

Jacobi–Abel addition theorem

11—20 of 265 matching pages

11: 1.15 Summability Methods

Abel Summability

Abel Means

Abel Summability

12: Errata

13: 18.10 Integral Representations

Jacobi

14: 22.6 Elementary Identities

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

15: 18.30 Associated OP’s

§18.30(i) Associated Jacobi Polynomials

Markov’s Theorem

16: 18.1 Notation

Classical OP’s

17: 30.10 Series and Integrals

18: 18.27 $q$ -Hahn Class

§18.27(iii) Big $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Jacobi

§18.27(iv) Little $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Little $q$ -Jacobi

From Little $q$ -Jacobi to Jacobi

19: 23.15 Definitions

20: 22.13 Derivatives and Differential Equations

Jacobi–Abel addition theorem

11—20 of 265 matching pages

11: 1.15 Summability Methods

Abel Summability

Abel Means

Abel Summability

12: Errata

13: 18.10 Integral Representations

Jacobi

14: 22.6 Elementary Identities

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

15: 18.30 Associated OP’s

§18.30(i) Associated Jacobi Polynomials

Markov’s Theorem

16: 18.1 Notation

Classical OP’s

17: 30.10 Series and Integrals

18: 18.27 q -Hahn Class

§18.27(iii) Big q -Jacobi Polynomials

From Big q -Jacobi to Jacobi

§18.27(iv) Little q -Jacobi Polynomials

From Big q -Jacobi to Little q -Jacobi

From Little q -Jacobi to Jacobi

19: 23.15 Definitions

20: 22.13 Derivatives and Differential Equations

18: 18.27 $q$ -Hahn Class

§18.27(iii) Big $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Jacobi

§18.27(iv) Little $q$ -Jacobi Polynomials

From Big $q$ -Jacobi to Little $q$ -Jacobi

From Little $q$ -Jacobi to Jacobi