DLMF: Untitled Document

… ►

35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

,

1\leq j\leq m

;

\|\mathbf{T}\|<1

.

ⓘ

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, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7 and Ch.35

… ►

Case $m=2$

►

35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

ⓘ

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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►Subject to the conditions (a)–(c), the function

f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)

is the unique solution of each partial differential equation ►

35.7.9

t_{j}(1-t_{j})\frac{{\partial}^{2}F}{{\partial t_{j}}^{2}}-\frac{1}{2}\sum_{% \begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{k}(1-t_{k})}{t_{j}-t_{k}}\frac{\partial F}{% \partial t_{k}}+\left({c-\tfrac{1}{2}(m-1)-\left(a+b-\tfrac{1}{2}(m-3)\right)t% _{j}}+\frac{1}{2}\sum_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{j}(1-t_{j})}{t_{j}-t_{k}}\right)\frac{% \partial F}{\partial t_{j}}=abF,

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer and $m$ : positive integer
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(iii), §35.7 and Ch.35

…

… ►Then (2.4.1) is valid in any closed sector with vertex

z=0

and properly interior to

-\alpha_{2}-\frac{1}{2}\pi<\operatorname{ph}z<-\alpha_{1}+\frac{1}{2}\pi

. … ►

(b)

$z$ ranges along a ray or over an annular sector $\theta_{1}\leq\theta\leq\theta_{2}$ , $|z|\geq Z$ , where $\theta=\operatorname{ph}z$ , $\theta_{2}-\theta_{1}<\pi$ , and $Z>0$ . $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$ .

… ►Higher coefficients

b_{2s}

in (2.4.15) can be found from (2.3.18) with

\lambda=1

,

\mu=2

, and

s

replaced by

2s

. …The last reference also includes examples, as do Olver (1997b, Chapter 4), Wong (1989, Chapter 2), and Bleistein and Handelsman (1975, Chapter 7). … ►with

a

and

b

chosen so that the zeros of

\ifrac{\partial p(\alpha,t)}{\partial t}

correspond to the zeros

w_{1}(\alpha),w_{2}(\alpha)

, say, of the quadratic

w^{2}+2aw+b

. …

… ►To code a message by this method, we replace each letter by two digits, say

A=01

,

B=02

,

\dots

,

Z=26

, and divide the message into pieces of convenient length smaller than the public value

n=pq

. …

… ►

Equations (18.16.12), (18.16.13)

18.16.12

(n+2)x_{n,1}\geq\left(n-1-\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1

18.16.13

(n+2)x_{n,n}\leq\left(n-1+\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1

The presentation of these inequalities has been improved.

… ►

Subsections 10.6(i), 10.29(i)

Sentences were added just below (10.6.5) and (10.29.3) regarding results on modified quotients of the form $\ifrac{z\mathscr{C}_{\nu\pm 1}\left(z\right)}{\mathscr{C}_{\nu}\left(z\right)}$ and $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ , respectively (suggested by Art Ballato on 2021-04-29).

… ►

Equations (22.9.8), (22.9.9) and (22.9.10)

22.9.8

s_{1,3}^{(4)}s_{2,3}^{(4)}+s_{2,3}^{(4)}s_{3,3}^{(4)}+s_{3,3}^{(4)}s_{1,3}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}}

22.9.9

c_{1,3}^{(4)}c_{2,3}^{(4)}+c_{2,3}^{(4)}c_{3,3}^{(4)}+c_{3,3}^{(4)}c_{1,3}^{(4% )}=-\frac{\kappa(\kappa+2)}{(1+\kappa)^{2}}

22.9.10

d_{1,3}^{(2)}d_{2,3}^{(2)}+d_{2,3}^{(2)}d_{3,3}^{(2)}+d_{3,3}^{(2)}d_{1,3}^{(2% )}=d_{1,3}^{(4)}d_{2,3}^{(4)}+d_{2,3}^{(4)}d_{3,3}^{(4)}+d_{3,3}^{(4)}d_{1,3}^% {(4)}=\kappa(\kappa+2)

Originally all the functions $s_{m,p}^{(4)}$ , $c_{m,p}^{(4)}$ , $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ in Equations (22.9.8), (22.9.9) and (22.9.10) were written incorrectly with $p=2$ . These functions have been corrected so that they are written with $p=3$ . In the sentence just below (22.9.10), the expression $s_{m,2}^{(4)}s_{n,2}^{(4)}$ has been corrected to read $s_{m,p}^{(4)}s_{n,p}^{(4)}$ .

Reported by Juan Miguel Nieto on 2019-11-07

… ►

Equations (28.28.21) and (28.28.22)

28.28.21

\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\cos\left((2\ell% +1)\phi\right)\operatorname{ce}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}A^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)

28.28.22

\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\operatorname{se}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h% \right),

Originally the prefactor $\frac{4}{\pi}$ and upper limit of integration $\pi/2$ in these two equations were given incorrectly as $\frac{2}{\pi}$ and $\pi$ .

Reported 2015-05-20 by Ruslan Kabasayev

… ►

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 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

.

► ►►►►►►

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
…
$\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
…
$\pi/4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
…
$2\pi/3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
$3\pi/4$	$\frac{1}{2}\sqrt{2}$	$-\frac{1}{2}\sqrt{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
…

► … ►

4.17.3

\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.17 and Ch.4

… ►If any lower argument in a

\mathit{6j}

symbol is

0

,

\tfrac{1}{2}

, or

1

, then the

\mathit{6j}

symbol has a simple algebraic form. … ►

34.5.5

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-% 2j_{3}+1)}{2j_{2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac% {1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.6

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j% _{3}+1)(J-2j_{3}+2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}% \right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.7

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}&j_{2}\end{Bmatrix}=(-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{% 1}(j_{1}+1))}{\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right% )^{\frac{1}{2}}}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

… ►

34.5.13

E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.

ⓘ

Defines:: $E(j)$ (locally)
Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

…

… ►The main functions treated in this chapter are the eigenvalues

a^{2m}_{\nu}\left(k^{2}\right)

,

a^{2m+1}_{\nu}\left(k^{2}\right)

,

b^{2m+1}_{\nu}\left(k^{2}\right)

,

b^{2m+2}_{\nu}\left(k^{2}\right)

, the Lamé functions

\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)

,

\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)

,

\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)

,

\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)

, and the Lamé polynomials

\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)

,

\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)

,

\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)

,

\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)

,

\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)

,

\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)

,

\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)

,

\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)

. … ►Other notations that have been used are as follows: Ince (1940a) interchanges

a^{2m+1}_{\nu}\left(k^{2}\right)

with

b^{2m+1}_{\nu}\left(k^{2}\right)

. The relation to the Lamé functions

L^{(m)}_{c\nu}

,

L^{(m)}_{s\nu}

of Jansen (1977) is given by … ►

\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=s_{\nu}^{2m+2}(k^{2}){\rm Es}_{% \nu}^{2m+2}(z,k^{2}),

►where the positive factors

c_{\nu}^{m}(k^{2})

and

s_{\nu}^{m}(k^{2})

are determined by …

… ►

Table 4.30.1: Hyperbolic functions: interrelations. …

► ►►►►►►►

	$\sinh\theta=a$	$\cosh\theta=a$	$\tanh\theta=a$	$\operatorname{csch}\theta=a$	$\operatorname{sech}\theta=a$	$\coth\theta=a$
$\sinh\theta$	$a$	$(a^{2}-1)^{1/2}$	$a(1-a^{2})^{-1/2}$	$a^{-1}$	$a^{-1}(1-a^{2})^{1/2}$	$(a^{2}-1)^{-1/2}$
$\cosh\theta$	$(1+a^{2})^{1/2}$	$a$	$(1-a^{2})^{-1/2}$	$a^{-1}(1+a^{2})^{1/2}$	$a^{-1}$	$a(a^{2}-1)^{-1/2}$
$\tanh\theta$	$a(1+a^{2})^{-1/2}$	$a^{-1}(a^{2}-1)^{1/2}$	$a$	$(1+a^{2})^{-1/2}$	$(1-a^{2})^{1/2}$	$a^{-1}$
$\operatorname{csch}\theta$	$a^{-1}$	$(a^{2}-1)^{-1/2}$	$a^{-1}(1-a^{2})^{1/2}$	$a$	$a(1-a^{2})^{-1/2}$	$(a^{2}-1)^{1/2}$
$\operatorname{sech}\theta$	$(1+a^{2})^{-1/2}$	$a^{-1}$	$(1-a^{2})^{1/2}$	$a(1+a^{2})^{-1/2}$	$a$	$a^{-1}(a^{2}-1)^{1/2}$
…

►

… ►

Polynomial $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$

… ►

Polynomial $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$

… ►

Polynomial $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$

… ►

Polynomial $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$

… ►

Polynomial $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$

…

… ►

§22.9(ii) Typical Identities of Rank 2

… ►These identities are cyclic in the sense that each of the indices

m, n

in the first product of, for example, the form

s_{m,p}^{(4)}s_{n,p}^{(4)}

are simultaneously permuted in the cyclic order:

m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1

;

n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1

. … ►

22.9.11

\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{2}d_% {1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),

ⓘ

Symbols:: $k^{\prime}$ : complementary modulus and $d_{m,p}^{(r)}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

►

22.9.12

c_{1,2}^{(2)}s_{1,2}^{(2)}d_{2,2}^{(2)}+c_{2,2}^{(2)}s_{2,2}^{(2)}d_{1,2}^{(2)% }=0.

ⓘ

Symbols:: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity and $d_{m,p}^{(r)}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

… ►

22.9.21

k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).

ⓘ

Symbols:: $k$ : modulus, $k^{\prime}$ : complementary modulus, $s_{m,p}^{(r)}$ : cyclic quantity and $c_{m,p}^{(r)}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

SL(2,Z)

31—40 of 803 matching pages

31: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Case $m=2$

32: 2.4 Contour Integrals

33: 27.16 Cryptography

34: Errata

35: 4.17 Special Values and Limits

36: 34.5 Basic Properties: $\mathit{6j}$ Symbol

37: 29.1 Special Notation

38: 4.30 Elementary Properties

39: 29.15 Fourier Series and Chebyshev Series

Polynomial $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$

Polynomial $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$

Polynomial $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$

Polynomial $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$

Polynomial $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$

40: 22.9 Cyclic Identities

§22.9(ii) Typical Identities of Rank 2

SL(2,Z)

31—40 of 803 matching pages

31: 35.7 Gaussian Hypergeometric Function of Matrix Argument

Case m = 2

32: 2.4 Contour Integrals

33: 27.16 Cryptography

34: Errata

35: 4.17 Special Values and Limits

36: 34.5 Basic Properties: 6 ⁢ j Symbol

37: 29.1 Special Notation

38: 4.30 Elementary Properties

39: 29.15 Fourier Series and Chebyshev Series

Polynomial 𝑢𝐸 2 ⁢ n m ⁡ ( z , k 2 )

Polynomial 𝑠𝐸 2 ⁢ n + 1 m ⁡ ( z , k 2 )

Polynomial 𝑠𝑐𝐸 2 ⁢ n + 2 m ⁡ ( z , k 2 )

Polynomial 𝑠𝑑𝐸 2 ⁢ n + 2 m ⁡ ( z , k 2 )

Polynomial 𝑐𝑑𝐸 2 ⁢ n + 2 m ⁡ ( z , k 2 )

40: 22.9 Cyclic Identities

§22.9(ii) Typical Identities of Rank 2

Case $m=2$

36: 34.5 Basic Properties: $\mathit{6j}$ Symbol

Polynomial $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$

Polynomial $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$

Polynomial $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$

Polynomial $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$

Polynomial $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$