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

► … ►See §22.17.

13: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►Corresponding expansions for

\theta_{j}'\left(z\middle|\tau\right)

j=1,2,3,4

, can be found by differentiating (20.2.1)–(20.2.4) with respect to

z

. … ►For fixed

\tau

, each

\theta_{j}\left(z\middle|\tau\right)

is an entire function of

z

with period

2\pi

;

\theta_{1}\left(z\middle|\tau\right)

is odd in

z

and the others are even. For fixed

z

, each of

\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}

\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}

\theta_{3}\left(z\middle|\tau\right)

, and

\theta_{4}\left(z\middle|\tau\right)

is an analytic function of

\tau

for

\Im\tau>0

, with a natural boundary

\Im\tau=0

, and correspondingly, an analytic function of

q

for

\left|q\right|<1

with a natural boundary

\left|q\right|=1

. … ►For

m,n\in\mathbb{Z}

, the

z

-zeros of

\theta_{j}\left(z\middle|\tau\right)

j=1,2,3,4

, are

(m+n\tau)\pi

(m+\tfrac{1}{2}+n\tau)\pi

(m+\tfrac{1}{2}+(n+\tfrac{1}{2})\tau)\pi

(m+(n+\tfrac{1}{2})\tau)\pi

respectively.

14: 15.17 Mathematical Applications

… ►

§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

15: 18.9 Recurrence Relations and Derivatives

… ►For

p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)

, … ►For

p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)

, … ►

Jacobi

… ►

Jacobi

… ►Further

n

-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). …

16: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.2

P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

;

\gamma,\delta,\nu\in\mathbb{C}

;

\Re\left(\gamma\right)>-1

ⓘ

Defines:: $P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Jacobi function of matrix argument
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

…

17: 27.9 Quadratic Characters

… ►If

p

does not divide

n

, then

(n|p)

has the value

1

when the quadratic congruence

x^{2}\equiv n\pmod{p}

has a solution, and the value

-1

when this congruence has no solution. … ►

27.9.3

(p|q)(q|p)=(-1)^{(p-1)(q-1)/4}.

ⓘ

Symbols:: $(\NVar{n}|\NVar{p})$ : Legendre symbol, $p$ : odd prime and $q$ : odd prime
Referenced by:: §27.9
Permalink:: http://dlmf.nist.gov/27.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.9 and Ch.27

►If an odd integer

P

has prime factorization

P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}

, then the Jacobi symbol

(n|P)

is defined by

(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}

, with

(n|1)=1

. The Jacobi symbol

(n|P)

is a Dirichlet character (mod

P

). …

18: Errata

… ►We have also incorporated material on continuous

q

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

Subsection 5.2(iii)

Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

… ►

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.

… ►

Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ( $\cosh z$ , instead of $\sinh z$ ).

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$

Reported 2010-11-23.

…

19: 18.18 Sums

… ►

Jacobi

… ►

Jacobi

… ►See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of

P^{(\gamma,\delta)}_{n}\left(x\right)

in terms of

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

. … ►

Jacobi

… ►See also (18.38.3) for a finite sum of Jacobi polynomials. …

20: 18.2 General Orthogonal Polynomials

… ►The constant function

p_{0}(x)

will often, but not always, be identically

1

(see, for example, (18.2.11_8)),

p_{-1}(x)=0

in all cases, by convention, as indicated in §18.1(i). … ►As in §18.1(i) we assume that

p_{-1}(x)\equiv 0

. … ►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. … … ►When the Jacobi matrix in (18.2.11_9) is truncated to an

n\times n

matrix …