"Koornwinder polynomials"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
* The Koornwinder polynomials, introduced in '''[Koornwinder92]''', are a generalisation of the [[Macdonald polynomials]] to the root system BCn. | * The Koornwinder polynomials, introduced in '''[Koornwinder92]''', are a generalisation of the [[Macdonald polynomials]] to the root system BCn. | ||
| − | ** | + | ** <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> |
| − | ** | + | ** <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]] | * 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 | + | * 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== | ==definition== | ||
;def (Koornwinder density) | ;def (Koornwinder density) | ||
| − | Throughout this section | + | Throughout this section <math>x=(x_1,\dots,x_n)</math>. |
Then the Koornwinder density is given by | Then the Koornwinder density is given by | ||
\begin{equation}\label{Eq_Kdensity} | \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}. | (x_ix_j,x_ix_j^{-1},x_i^{-1}x_j,x_i^{-1}x_j^{-1};q)_{\infty}. | ||
\end{align*} | \end{align*} | ||
| − | For complex | + | For complex <math>q,t,t_0,\dots,t_3</math> such that |
| − | + | <math>\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1</math> this | |
| − | defines a scalar product on | + | defines a scalar product on <math>\mathbb{C}[x^{\pm 1}]</math> via |
\[ | \[ | ||
\langle{f}, {g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= | \langle{f}, {g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= | ||
| 36번째 줄: | 36번째 줄: | ||
\frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. | \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. | ||
\] | \] | ||
| − | Let | + | Let <math>W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n</math> be the hyperoctahedral group |
| − | with natural action on | + | with natural action on <math>\mathbb{C}[x^{\pm}]</math>. For <math>\lambda</math> a partition |
| − | of length at most | + | of length at most <math>n</math>, let <math>m_{\lambda}^W</math> be the <math>W</math>-invariant monomial |
symmetric function | symmetric function | ||
\[ | \[ | ||
m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} | m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} | ||
\] | \] | ||
| − | summed over all | + | summed over all <math>\alpha</math> in the <math>W</math>-orbit of <math>\lambda</math>. |
;def (Koornwinder polynomial) | ;def (Koornwinder polynomial) | ||
In analogy with the Macdonald polynomials, the Koornwinder | In analogy with the Macdonald polynomials, the Koornwinder | ||
| − | polynomials | + | 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 | are defined as the unique family of polynomials in | ||
| − | + | <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} | K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} | ||
| 59번째 줄: | 59번째 줄: | ||
===notation=== | ===notation=== | ||
| − | + | :<math> | |
\newcommand{\la}{\lambda} | \newcommand{\la}{\lambda} | ||
\newcommand{\La}{\Lambda} | \newcommand{\La}{\Lambda} | ||
| 68번째 줄: | 68번째 줄: | ||
\newcommand{\C}{\mathrm C} | \newcommand{\C}{\mathrm C} | ||
\newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} | \newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} | ||
| − | + | </math> | |
==properties== | ==properties== | ||
| − | * From the definition it follows that the | + | * From the definition it follows that the <math>K_{\lambda}</math> are symmetric under permutation of the <math>t_r</math>. |
===quadratic norm=== | ===quadratic norm=== | ||
The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) | The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) | ||
| 86번째 줄: | 86번째 줄: | ||
===Cauchy identity=== | ===Cauchy identity=== | ||
;thm (Mimachi, '''[Mimachi01]''' thm 2.1) | ;thm (Mimachi, '''[Mimachi01]''' thm 2.1) | ||
| − | The | + | The <math>\mathrm{BC}_n</math> analogue of the Cauchy identity |
is given by | is given by | ||
\begin{align}\label{Eq_Mim} | \begin{align}\label{Eq_Mim} | ||
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&=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), | &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), | ||
\end{align} | \end{align} | ||
| − | where | + | where <math>y=(y_1,\dots,y_m)</math> and <math>(a-b^{\pm}):=(a-b)(a-b^{-1})</math>. |
| + | |||
| + | |||
| + | ===relation with Macdonald polynomials=== | ||
| + | * <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> | ||
| + | * <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> | ||
| + | * <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> | ||
| + | * <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> | ||
==expression in terms of interpolation polynomials== | ==expression in terms of interpolation polynomials== | ||
* binomial coefficient | * binomial coefficient | ||
| − | + | \[ | |
{\lambda \brack \mu}_{q,t,s} | {\lambda \brack \mu}_{q,t,s} | ||
:= | := | ||
\frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})} | \frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})} | ||
{\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})} | {\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})} | ||
| − | + | \] | |
* Okounkov's binomial formula (Theorem 7.10, '''[Okounkov98]''') gives an expansion of the Koornwinder polynomials in terms of [[BCn interpolation polynomials]] | * 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 | + | * 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} | K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} | ||
{\lambda \brack \mu}_{q,t,s} \, | {\lambda \brack \mu}_{q,t,s} \, | ||
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{K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, | {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), | \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), | ||
| − | + | </math> | |
| − | where | + | where <math>s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}</math>. |
| 143번째 줄: | 150번째 줄: | ||
==history== | ==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 | + | * 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. | * 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. | ||
| 179번째 줄: | 186번째 줄: | ||
[[분류:symmetric polynomials]] | [[분류:symmetric polynomials]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q6430769 Q6430769] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'koornwinder'}, {'LEMMA': 'polynomial'}] | ||
2021년 2월 17일 (수) 01:32 기준 최신판
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 \((q;a,b,c,d)\)
- \(n \geq 2\), they depend on six parameters \((q,t;t_0,t_1,t_2,t_3)\)
- 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 \((C^{\vee}_n,C_n)\)
definition
- def (Koornwinder density)
Throughout this section \(x=(x_1,\dots,x_n)\). Then the Koornwinder density is given by \begin{equation}\label{Eq_Kdensity} \Delta(x;q,t;t_0,t_1,t_2,t_3):= \prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} {\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} \prod_{1\leq i<j\leq n} \frac{(x_i^{\pm}x_j^{\pm};q)_{\infty}} {(tx_i^{\pm}x_j^{\pm};q)_{\infty}}, \end{equation} where \begin{align*} (x_i^{\pm};q)_{\infty}&:=(x_i,x_i^{-1};q)_{\infty} \\ (x_i^{\pm}x_j^{\pm};q)_{\infty}&:= (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 \(\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1\) this defines a scalar product on \(\mathbb{C}[x^{\pm 1}]\) via \[ \langle{f}, {g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;t_0,t_1,t_2,t_3) \,d T(x), \] where \[ d T(x):=\frac{1}{2^n n! (2\pi i)^n}\, \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. \] Let \(W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n\) be the hyperoctahedral group with natural action on \(\mathbb{C}[x^{\pm}]\). For \(\lambda\) a partition of length at most \(n\), let \(m_{\lambda}^W\) be the \(W\)-invariant monomial symmetric function \[ m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} \] summed over all \(\alpha\) in the \(W\)-orbit of \(\lambda\).
- def (Koornwinder polynomial)
In analogy with the Macdonald polynomials, the Koornwinder polynomials \(K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)\) are defined as the unique family of polynomials in \(\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W\) such that [43] \[ K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} \] and \begin{equation}\label{Eq_KKnul} \langle K_{\lambda}, K_{\mu}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =0 \qquad\text{if }\lambda\neq\mu. \end{equation}
notation
\[ \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\tees}{t_0,t_1,t_2,t_3} \newcommand{\B}{\mathrm B} \newcommand{\BC}{\mathrm{BC}} \newcommand{\C}{\mathrm C} \newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} \]
properties
- From the definition it follows that the \(K_{\lambda}\) are symmetric under permutation of the \(t_r\).
quadratic norm
The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) and \cite{Sahi99} (general case). For our purposes we only need \begin{equation}\label{Eq_Gus} \langle{1},{1}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =\prod_{i=1}^n \frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} {(q,t^i;q)_{\infty}\prod_{0\leq r<s\leq 3}(t_rt_st^{i-1};q)_{\infty}}, \end{equation} known as Gustafson's integral [Gustafson90]
Cauchy identity
- thm (Mimachi, [Mimachi01] thm 2.1)
The \(\mathrm{BC}_n\) analogue of the Cauchy identity is given by \begin{align}\label{Eq_Mim} \sum_{\lambda\subseteq m^n} (-1)^{\lvert{\lambda}\rvert} K_{m^n-\lambda}(x;q,t;t_0,t_1,t_2,t_3) K_{\lambda'}(y;t,q;t_0,t_1,t_2,t_3) \\ &=\prod_{i=1}^n\prod_{j=1}^m \big(x_i+x_i^{-1}-y_j-y_j^{-1}\big)\\ &=\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 \((a-b^{\pm}):=(a-b)(a-b^{-1})\).
relation with Macdonald polynomials
- \(P_{\lambda}^{(B_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2^{1/2},q^{1/2})\)
- \(P_{\lambda}^{(B_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2, t_2 q^{1/2})\)
- \(P_{\lambda}^{(C_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;q^{1/2},-q^{1/2},t_2^{1/2},-t_2^{1/2})\)
- \(P_{\lambda}^{(C_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;(t_2q)^{1/2},-(t_2q)^{1/2},t_2^{1/2},-t_2^{1/2})\)
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:
\[ K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} {\lambda \brack \mu}_{q,t,s} \, \frac{K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)} {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), \] where \(s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}\).
Now let \cite{Rains05}
\[
C_{\lambda}^{+}(z;q,t):=\prod_{(i,j)\in \lambda}
\big(1-zq^{\lambda_i+j-1}t^{2-\lambda'_j-i}\big).
\]
By \cite[Proposition 4.1]{Rains05}
\[
\qbin{m^n}{\mu}_{q,t,s}=(-q)^{\abs{\mu}} t^{n(\mu)} q^{n(\mu')}\,
\frac{(t^n,q^{-m},s^2q^mt^{1-n};q,t)_{\mu}}
{C_{\mu}^{-}(q,t;q,t)C_{\mu}^{+}(s^2;q,t)}
\]
and the specialisation formulas \cite{vDiejen96,Sahi99}
\begin{multline*}
K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;\tees\big) \\
=\frac{t^{n(\lambda)}}{(t_0t^{n-1})^{\abs{\la}}} \cdot
\frac{(t^n,t_0t_1t^{n-1},t_0t_2t^{n-1},t_0t_3t^{n-1};q,t)_{\la}}
{C^{-}_{\la}(t;q,t)C^{+}_{\la}(t_0t_1t_2t_3t^{2n-2}/q;q,t)}
\end{multline*}
and \cite[Corollary 3.11]{Rains05}
\[
\bar{P}_{\mu}^{\ast}\big(z(1,t,\dots,t^{n-1});q,t,s\big)
=\frac{t^{2n(\mu)} q^{-n(\mu')}}{(-st^{n-1})^{\abs{\mu}}}\cdot
\frac{(t^n,s/z,szt^{n-1};q,t)_{\mu}} {C_{\mu}^{-}(t;q,t)}
\]
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 \(BC_n\)
- 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.
- Askey-Wilson polynomials
- Gustafson integral
- BCn interpolation polynomials
- Littlewood identities for Macdonald polynomials and PCnBn
encyclopedia
expositions
- Rains, Eric M. "Elliptic Analogues of the Macdonald and Koornwinder Polynomials." Proceedings of the International Congress of Mathematicians. Vol. 901. No. 2010. 2010. http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.2530.2554.pdf
- 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, Koornwinder-Macdonald Polynomials
articles
- van Diejen, J. F., and E. Emsiz. “Branching Rules for Symmetric Hypergeometric Polynomials.” arXiv:1601.06186 [math-Ph], January 22, 2016. http://arxiv.org/abs/1601.06186.
- Corteel, Sylvie, and Lauren Williams. ‘Macdonald-Koornwinder Moments and the Two-Species Exclusion Process’. arXiv:1505.00843 [cond-Mat, Physics:nlin], 4 May 2015. http://arxiv.org/abs/1505.00843.
- Stokman, Jasper, and Bart Vlaar. “Koornwinder Polynomials and the XXZ Spin Chain.” Journal of Approximation Theory 197 (September 2015): 69–100. doi:10.1016/j.jat.2014.03.003.
- van Diejen, J. F., and E. Emsiz. “Branching Formula for Macdonald-Koornwinder Polynomials.” arXiv:1408.2280 [math], August 10, 2014. http://arxiv.org/abs/1408.2280.
- Rains, Eric M. “BCn-Symmetric Polynomials.” Transformation Groups 10, no. 1 (March 2005): 63–132. doi:10.1007/s00031-005-1003-y. http://arxiv.org/abs/math/0112035.
- [Mimachi01] K. Mimachi, A duality of Macdonald--Koornwinder polynomials and its application to integral representations, Duke Math. J.107 (2001), 265--281.
- Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
- S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267--282.
- [Okounkov98] * Okounkov, A. “BC-Type Interpolation Macdonald Polynomials and Binomial Formula for Koornwinder Polynomials.” Transformation Groups 3, no. 2 (June 1998): 181–207. doi:10.1007/BF01236432.
- J. F. van Diejen, Self-dual Koornwinder--Macdonald polynomials, Invent. Math. 126 (1996), 319--339.
- van Diejen, J. F. “Commuting Difference Operators with Polynomial Eigenfunctions.” arXiv:funct-an/9306002, June 7, 1993. http://arxiv.org/abs/funct-an/9306002.
- [Koornwinder92] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, pp. 189–204, Contemp. Math. 138, Amer. Math. Soc., Providence, 1992. http://oai.cwi.nl/oai/asset/2292/2292A.pdf
- [Gustafson90] R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97--105.
메타데이터
위키데이터
- ID : Q6430769
Spacy 패턴 목록
- [{'LOWER': 'koornwinder'}, {'LEMMA': 'polynomial'}]