Koornwinder polynomials - 편집 역사

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Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T02:35:51Z

메타데이터: 새 문단

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imported>Pythagoras0: /* expression in terms of interpolation polynomials */

2020-03-18T05:40:40Z

expression in terms of interpolation polynomials

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imported>Pythagoras0: /* expositions */

2017-05-26T07:20:44Z

expositions

imported>Pythagoras0: /* expositions */

2017-05-26T07:17:19Z

expositions

imported>Pythagoras0: /* notation */

2017-05-17T02:20:29Z

notation

@@ 188번째 줄: / 188번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q6430769 Q6430769]

← 이전 판		2020년 12월 28일 (월) 02:35 판
187번째 줄:		187번째 줄:
	[[분류:symmetric polynomials]]		[[분류:symmetric polynomials]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q6430769 Q6430769]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * The Koornwinder polynomials, introduced in '''[Koornwinder92]''',  are a generalisation of the [[Macdonald polynomials]] to the root system BCn.
-** $n = 1$, they correspond to the [[Askey-Wilson polynomials]] depending on 5-parameter $(q;t_0,t_1,t_2,t_3)$ or $(q;a,b,c,d)$
+** <math>n = 1</math>, they correspond to the [[Askey-Wilson polynomials]] depending on 5-parameter <math>(q;t_0,t_1,t_2,t_3)</math> or <math>(q;a,b,c,d)</math>
-** $n \geq 2$, they depend on six parameters $(q,t;t_0,t_1,t_2,t_3)$
+** <math>n \geq 2</math>, they depend on six parameters <math>(q,t;t_0,t_1,t_2,t_3)</math>
 * It is known that all Macdonald polynomials associated with the classical (nonexceptional) root systems can be considered as their special cases, see [[Macdonald polynomials associatied with root systems]]
-* In Macdonald's theory of orthogonal polynomials associated to root systems, the Koornwinder-Macdonald polynomials is a family which corresponds to the non-reduced irreducible affine root system of type $(C^{\vee}_n,C_n)$
+* In Macdonald's theory of orthogonal polynomials associated to root systems, the Koornwinder-Macdonald polynomials is a family which corresponds to the non-reduced irreducible affine root system of type <math>(C^{\vee}_n,C_n)</math>
 ==definition==
 ;def (Koornwinder density)
-Throughout this section $x=(x_1,\dots,x_n)$.
+Throughout this section <math>x=(x_1,\dots,x_n)</math>.
 Then the Koornwinder density is given by
 \begin{equation}\label{Eq_Kdensity}
@@ 24번째 줄: / 24번째 줄: @@
 (x_ix_j,x_ix_j^{-1},x_i^{-1}x_j,x_i^{-1}x_j^{-1};q)_{\infty}.
 \end{align*}
-For complex $q,t,t_0,\dots,t_3$ such that
+For complex <math>q,t,t_0,\dots,t_3</math> such that
-$\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1$ this
+<math>\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1</math> this
-defines a scalar product on $\mathbb{C}[x^{\pm 1}]$ via
+defines a scalar product on <math>\mathbb{C}[x^{\pm 1}]</math> via
 \[
 \langle{f}, {g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:=
@@ 36번째 줄: / 36번째 줄: @@
 \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}.
 \]
-Let $W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group
+Let <math>W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n</math> be the hyperoctahedral group
-with natural action on $\mathbb{C}[x^{\pm}]$. For $\lambda$ a partition
+with natural action on <math>\mathbb{C}[x^{\pm}]</math>. For <math>\lambda</math> a partition
-of length at most $n$, let $m_{\lambda}^W$ be the $W$-invariant monomial
+of length at most <math>n</math>, let <math>m_{\lambda}^W</math> be the <math>W</math>-invariant monomial
 symmetric function
 \[
 m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha}
 \]
-summed over all $\alpha$ in the $W$-orbit of $\lambda$.
+summed over all <math>\alpha</math> in the <math>W</math>-orbit of <math>\lambda</math>.
 ;def (Koornwinder polynomial)
 In analogy with the Macdonald polynomials, the Koornwinder
-polynomials $K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)$
+polynomials <math>K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)</math>
 are defined as the unique family of polynomials in
-$\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W$ such that [43]
+<math>\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W</math> such that [43]
 \[
 K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu}
@@ 59번째 줄: / 59번째 줄: @@
 ===notation===
-$$
+:<math>
 \newcommand{\la}{\lambda}
 \newcommand{\La}{\Lambda}
@@ 68번째 줄: / 68번째 줄: @@
 \newcommand{\C}{\mathrm C}
 \newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}
-$$
+</math>
 ==properties==
-* From the definition it follows that the $K_{\lambda}$ are symmetric under permutation of the $t_r$.
+* From the definition it follows that the <math>K_{\lambda}</math> are symmetric under permutation of the <math>t_r</math>.
 ===quadratic norm===
 The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case)
@@ 86번째 줄: / 86번째 줄: @@
 ===Cauchy identity===
 ;thm (Mimachi, '''[Mimachi01]''' thm 2.1)
-The $\mathrm{BC}_n$ analogue of the Cauchy identity
+The <math>\mathrm{BC}_n</math> analogue of the Cauchy identity
 is given by
 \begin{align}\label{Eq_Mim}
@@ 94번째 줄: / 94번째 줄: @@
 &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big),
 \end{align}
-where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.
+where <math>y=(y_1,\dots,y_m)</math> and <math>(a-b^{\pm}):=(a-b)(a-b^{-1})</math>.
 ===relation with Macdonald polynomials===
-* $P_{\lambda}^{(B_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2^{1/2},q^{1/2})$
+* <math>P_{\lambda}^{(B_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2^{1/2},q^{1/2})</math>
-* $P_{\lambda}^{(B_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2, t_2 q^{1/2})$
+* <math>P_{\lambda}^{(B_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2, t_2 q^{1/2})</math>
-* $P_{\lambda}^{(C_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;q^{1/2},-q^{1/2},t_2^{1/2},-t_2^{1/2})$
+* <math>P_{\lambda}^{(C_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;q^{1/2},-q^{1/2},t_2^{1/2},-t_2^{1/2})</math>
-* $P_{\lambda}^{(C_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;(t_2q)^{1/2},-(t_2q)^{1/2},t_2^{1/2},-t_2^{1/2})$
+* <math>P_{\lambda}^{(C_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;(t_2q)^{1/2},-(t_2q)^{1/2},t_2^{1/2},-t_2^{1/2})</math>
@@ 113번째 줄: / 113번째 줄: @@
 \]
 * Okounkov's binomial formula (Theorem 7.10, '''[Okounkov98]''') gives an expansion of the Koornwinder polynomials in terms of [[BCn interpolation polynomials]]
-* The coefficients in this expansion are given by $BC_n$ $q$-binomial coefficients ${\lambda \brack \mu}_{q,t,s}$ times a ratio of principally specialised Koornwinder polynomials:
+* The coefficients in this expansion are given by <math>BC_n</math> <math>q</math>-binomial coefficients <math>{\lambda \brack \mu}_{q,t,s}</math> times a ratio of principally specialised Koornwinder polynomials:
-$$
+:<math>
 K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda}
 {\lambda \brack \mu}_{q,t,s} \,
@@ 120번째 줄: / 120번째 줄: @@
 {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\,
 \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0),
-$$
+</math>
-where $s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}$.
+where <math>s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}</math>.
@@ 150번째 줄: / 150번째 줄: @@
 ==history==
-* Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system $BC_n$
+* Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system <math>BC_n</math>
 * The various families of Macdonald (orthogonal) polynomials for classical root systems are all contained in the Koornwinder polynomials, and for a long time it was assumed they represented the highest possible level of generalisation.

← 이전 판		2020년 11월 13일 (금) 16:35 판
186번째 줄:		186번째 줄:

	[[분류:symmetric polynomials]]		[[분류:symmetric polynomials]]
		+	[[분류:migrate]]

@@ 106번째 줄: / 106번째 줄: @@
 ==expression in terms of interpolation polynomials==
 * binomial coefficient
-$$
+\[
 {\lambda \brack \mu}_{q,t,s}
 :=
 \frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})}
 {\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})}
-$$
+\]
 * Okounkov's binomial formula (Theorem 7.10, '''[Okounkov98]''') gives an expansion of the Koornwinder polynomials in terms of [[BCn interpolation polynomials]]
 * The coefficients in this expansion are given by $BC_n$ $q$-binomial coefficients ${\lambda \brack \mu}_{q,t,s}$ times a ratio of principally specialised Koornwinder polynomials:

← 이전 판		2020년 3월 13일 (금) 06:28 판
95번째 줄:		95번째 줄:
	\end{align}		\end{align}
	where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.		where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.
		+
		+
		+	===relation with Macdonald polynomials===
		+	* $P_{\lambda}^{(B_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2^{1/2},q^{1/2})$
		+	* $P_{\lambda}^{(B_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2, t_2 q^{1/2})$
		+	* $P_{\lambda}^{(C_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;q^{1/2},-q^{1/2},t_2^{1/2},-t_2^{1/2})$
		+	* $P_{\lambda}^{(C_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;(t_2q)^{1/2},-(t_2q)^{1/2},t_2^{1/2},-t_2^{1/2})$

@@ 161번째 줄: / 161번째 줄: @@
 * Stokman, Jasper V. “Lecture Notes on Koornwinder Polynomials.” In Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207. Adv. Theory Spec. Funct. Orthogonal Polynomials. Nova Sci. Publ., Hauppauge, NY, 2004. http://www.ams.org/mathscinet-getitem?mr=2085855.
 * Stokman, Jasper V. “Macdonald-Koornwinder Polynomials.” arXiv:1111.6112 [math], November 25, 2011. http://arxiv.org/abs/1111.6112.
-* Stokman, [https://math.la.asu.edu/~sf2000/stokman_ex.pdf Macdonald-Koornwinder Polynomials]
+* Stokman, [https://math.la.asu.edu/~sf2000/stokman_ex.pdf Koornwinder-Macdonald Polynomials]
 ==articles==

@@ 67번째 줄: / 67번째 줄: @@
 \newcommand{\BC}{\mathrm{BC}}
 \newcommand{\C}{\mathrm C}
+$