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2: 2.2 Transcendental Equations

… ►Then for

y>f(a)

the equation

f(x)=y

has a unique root

x=x(y)

(a,\infty)

, and ►

2.2.2

x(y)\sim y,

y\to\infty

ⓘ

Symbols:: $\sim$ : asymptotic equality and $y$ : root
Referenced by:: §2.2
Permalink:: http://dlmf.nist.gov/2.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2 and Ch.2

… ►

2.2.3

t^{2}-\ln t=y.

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.4

t=y^{\frac{1}{2}}\left(1+o\left(1\right)\right),

y\to\infty

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

… ►

2.2.5

t^{2}=y+\ln t=y+\tfrac{1}{2}\ln y+o\left(1\right),

ⓘ

Symbols:: $o\left(\NVar{x}\right)$ : order less than, $\ln\NVar{z}$ : principal branch of logarithm function and $y$ : root
Permalink:: http://dlmf.nist.gov/2.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

…

3: 37.9 Jacobi Polynomials Associated with Root System $A_{2}$

§37.9 Jacobi Polynomials Associated with Root System $A_{2}$

… ►For

\alpha>-\frac{5}{6}

Jacobi polynomials associated with root system $A_{2}$ are polynomials

P_{m,n}^{\alpha}(z,\overline{z})

(

m,n\in\mathbb{N}_{0}

) which are defined uniquely, up to constant factors, by … ►These reflections generate a transformation group of the plane

t_{1}+t_{2}+t_{3}=0

which contains as a subgroup the Weyl group

W

for root system

A_{2}

. The group

W

is isomorphic to the symmetric group

S_{3}

and permutes

t_{1}-t_{2},t_{2}-t_{3},t_{3}-t_{1}

. … ►The polynomials

P_{m,n}^{\pm\frac{1}{2}}(z,\overline{z})

are called Chebyshev polynomials of the first and the second kind for root system

A_{2}

. …

4: 1.11 Zeros of Polynomials

… ►Roots of

f(z)=0

are

2+\sqrt[3]{4}+\sqrt[3]{2}

2+\sqrt[3]{4}\rho+\sqrt[3]{2}\rho^{2}

2+\sqrt[3]{4}\rho^{2}+\sqrt[3]{2}\rho

. … ►The square roots are chosen so that … ►

§1.11(iv) Roots of Unity and of Other Constants

►The roots of … ►The roots of …

5: 37.8 Jacobi Polynomials Associated with Root System $BC_{2}$

§37.8 Jacobi Polynomials Associated with Root System $BC_{2}$

… ►the Jacobi polynomials associated with root system $BC_{2}$ are symmetric polynomials

p_{k,n}^{\alpha,\beta,\gamma}(x,y)

(

0\leq k\leq n

) of the form … ►

37.8.5

L_{\alpha,\beta,\gamma}{}p_{k,n}^{\alpha,\beta,\gamma}=\lambda_{k,n}p_{k,n}^{% \alpha,\beta,\gamma},

ⓘ

Symbols:: $n$ : nonnegative integer, $k$ : nonnegative integer, $\lambda_{k,n}$ : eignvalue for two-variable $p$ Jacobi polynomial associated with root system $BC_{2}$ , $\alpha$ : real parameter, $\beta$ : real parameter, $\gamma$ : real parameter, $p_{k,n}^{\alpha,\beta,\gamma}$ : two-variable $p$ Jacobi polynomial associated with root system $BC_{2}$ and $L_{\alpha,\beta,\gamma}$ : differential operator for two-variable $p$ Jacobi polynomial associated with root system $BC_{2}$
Proved:: Koornwinder (1974b, (4.4))(proved)
Permalink:: http://dlmf.nist.gov/37.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §37.8, §37.8 and Ch.37

… ►Moreover, the corresponding OPs

P_{k,n}(u,v)

as in (37.8.11) satisfy for

\gamma=\pm\frac{1}{2}

the property that

\{P_{k,n}\}_{k=0}^{n}

has

\tfrac{1}{2}(n+1)(n+2)

real common zeros; see Schmid and Xu (1994). … ►on the square, the OPs are related to the Jacobi polynomials for root system

BC_{2}

. …

6: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►Let

G

be a reflection group (also called Coxeter group) with reduced root system

R

(see Dunkl and Xu (2014, §6.2) for definitions of these notions). Let

R_{+}

be the set of positive roots and let

\mathbf{v}\mapsto\kappa_{\mathbf{v}}

be a nonnegative function defined on

R_{+}

with the property that it takes constant value in each conjugacy class of roots. … ►

§37.19(vi) OPs Associated with Root Systems

… ►For general

q

they occur as Macdonald polynomials for root system $A_{n}$ , as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992). …

7: 23.7 Quarter Periods

… ►

23.7.1

\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.2

\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

23.7.3

\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►where

k,k^{\prime}

and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

8: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

9: 23.21 Physical Applications

… ►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation ►

23.21.1

\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $z$ : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

… ►

23.21.3

f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

►Another form is obtained by identifying

e_{1}

e_{2}

e_{3}

as lattice roots (§23.3(i)), and setting …

10: 23.3 Differential Equations

… ►

§23.3(i) Invariants, Roots, and Discriminant

… ►The lattice roots satisfy the cubic equation … ►

23.3.4

\Delta={g_{2}}^{3}-27{g_{3}}^{2}=16(e_{2}-e_{3})^{2}(e_{3}-e_{1})^{2}(e_{1}-e_% {2})^{2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $\Delta$ : discriminant
A&S Ref:: 18.1.8
Referenced by:: §23.19
Permalink:: http://dlmf.nist.gov/23.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.5

e_{1}+e_{2}+e_{3}=0,

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.11
Referenced by:: (23.22.1), §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

… ►

23.3.7

g_{3}=4e_{1}e_{2}e_{3}=\tfrac{4}{3}({e_{1}}^{3}+{e_{2}}^{3}+{e_{3}}^{3}).

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ and $\mathbb{L}$ : lattice
A&S Ref:: 18.1.10
Referenced by:: (23.22.1), §23.22(ii), §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §23.3(i), §23.3 and Ch.23

…