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| In mathematics, the '''Dyson conjecture''' {{harvs |first=Freeman |last=Dyson |year=1962 |authorlink=Freeman Dyson}} is a conjecture about the constant term of certain [[Laurent polynomial]]s, proved by [[Kenneth G. Wilson|Wilson]] and Gunson. Andrews generalized it to the '''q-Dyson conjecture''', proved by Zeilberger and Bressoud and sometimes called the '''Zeilberger–Bressoud theorem'''. Macdonald generalized it further to more general root systems with the '''Macdonald constant term conjecture''', proved by Cherednik.
| | I'm a 33 years old, married and study at the high school (Educational Policy Studies).<br>In my spare time I teach myself German. I've been twicethere and look forward to returning anytime soon. I love to read, preferably on my kindle. I really love to watch The Big Bang Theory and 2 Broke Girls as well as docus about anything geological. I like Stone collecting.<br>xunjie 国内の人々に影響を与える。 |
| | パーソナライズされた熟女との完全な衣装のシャツを表示す�<br> |
| | デザイナーのクリストフ·ルメールチップの材料にさらに注意を比較した。 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����ߥ�� �rӋ ���] バングラデシュの衣服の輸出は130億ドルのターゲットバングラデシュの4,000人以上の縫製工場が、 |
| | マグレディ分析ボーアファッション有限調印式のために演説を行った調印式の意義は成都テキスタイル·カレッジ·レディ·ボーア研究所だけでなく、 |
| | 今シーズンの完全な焦点になりたい？今シーズンの販売マスターになりたい？ディWeinaはすぐ割引ブランドの仲間入りをし、 [http://www.tobler-verlag.ch/media/galerie/jp/shop/nb/ �˥�`�Х�� �����ȥ�å�] 次の確立ZigNanoで実行するための動きの異なるオプションを言及するとバスケットボール世界ZigEncore後、 |
| | その素朴な雰囲気のバッグ本体ラインと強力な機能を保持し、 |
| | チャンツィイーとSaが恋にBeiningエンターテインメントゴシップファン心配、[http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����ߥ�� �rӋ] 美しいレースが大きな金属バックルガードル著名なスタイル、 |
| | 外の美しさの気質のバージョンは、 |
| | キャットアイサングラスの1組のシンプルな黒のパンツ、 |
| | 財布やiPhoneケースを含む単一の特殊な製品の彼の英国風の創作を販売しています。 [http://belpars.by/Files/Config/tp/new/best/toms.html �ȥॺ ����<br><br> ���å�] |
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| ==Dyson conjecture==
| | my website; [http://www.steadfast-hawaii.org/img/p/hot/chloe.php クロエ 財布] |
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| The Dyson conjecture states that the [[Laurent polynomial]]
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| :<math>\prod _{1\le i\ne j\le n}(1-t_i/t_j)^{a_i}</math>
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| has constant term
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| :<math>\frac{(a_1+a_2+\cdots+a_n)!}{a_1!a_2!\cdots a_n!}.</math>
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| The conjecture was first proved independently by {{harvtxt|Wilson|1962}} and {{harvtxt|Gunson|1962}}. {{harvtxt|Good|1970}} later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations
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| :<math>F(a_1,\dots,a_n) = \sum_{i=1}^nF(a_1,\dots,a_i-1,\dots,a_n).</math>
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| The case ''n'' = 3 of Dyson's conjecture follows from the [[Dixon identity]].
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| {{harvtxt|Sills|Zeilberger|2006}} and {{harv|Sills|2006}} used a computer to find expressions for non-constant coefficients of
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| Dyson's Laurent polynomial.
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| ==Dyson integral==
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| When all the values ''a''<sub>''i''</sub> are equal to β/2, the constant term in Dyson's conjecture is the value of '''Dyson's integral'''
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| :<math>\frac{1}{(2\pi)^n}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{1\le j<k\le n}|e^{i\theta_j}-e^{i\theta_k}|^\beta \, d\theta_1\cdots d\theta_n.</math> | |
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| Dyson's integral is a special case of [[Selberg's integral]] after a change of variable and has value
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| :<math>\frac{\Gamma(1+\beta n/2)}{\Gamma(1+\beta/2)^n}</math>
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| which gives another proof of Dyson's conjecture in this special case.
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| ==''q''-Dyson conjecture==
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| {{harvtxt|Andrews|1975}} found a [[q-analog]] of Dyson's conjecture, stating that the constant term of
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| :<math>\prod_{1\le i<j\le n}\left(\frac{x_i}{x_j};q\right)_{a_i}\left(\frac{qx_j}{x_i};q\right)_{a_j}</math>
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| is
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| :<math>\frac{(q;q)_{a_1+\cdots+a_n}}{(q;q)_{a_1}\cdots(q;q)_{a_n}}.</math>
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| Here (''a'';''q'')<sub>''n''</sub> is the [[q-Pochhammer symbol]].
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| This conjecture reduces to Dyson's conjecture for ''q''=1, and was proved by {{harvtxt|Zeilberger|Bressoud|1985}}.
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| ==Macdonald conjectures==
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| {{harvtxt|Macdonald|1982}} extended the conjecture to arbitrary finite or affine [[root system]]s, with Dyson's original conjecture corresponding to
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| the case of the ''A''<sub>''n''−1</sub> root system and Andrews's conjecture corresponding to the affine ''A''<sub>''n''−1</sub> root system. Macdonald reformulated these conjectures as conjectures about the norms of [[Macdonald polynomial]]s. Macdonald's conjectures were proved by {{harv|Cherednik|1995}} using doubly affine Hecke algebras.
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| [[Ian G. Macdonald|Macdonald]]'s form of Dyson's conjecture for root systems of type BC is closely related to [[Selberg's integral]].
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| ==References==
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| *{{Citation | authorlink=George Andrews (mathematician) | last1=Andrews | first1=George E. | title=Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) | publisher=[[Academic Press]] | location=Boston, MA | mr=0399528 | year=1975 | chapter=Problems and prospects for basic hypergeometric functions | pages=191–224}}
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| *{{citation |jstor=2118632 |pages=191–216 |last1=Cherednik |first1=I. |title=Double Affine Hecke Algebras and Macdonald's Conjectures |volume=141 |issue=1 |journal=The Annals of Mathematics |year=1995 |doi=10.2307/2118632}}
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| *{{Citation | last1=Dyson | first1=Freeman J. | title=Statistical theory of the energy levels of complex systems. I | doi=10.1063/1.1703773 | mr=0143556 | year=1962 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=3 | pages=140–156}}
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| *{{Citation | last1=Good | first1=I. J. | author1-link=I. J. Good | title=Short proof of a conjecture by Dyson | doi=10.1063/1.1665339 | mr=0258644 | year=1970 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=11 | pages=1884 | issue=6}}
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| *{{Citation | last1=Gunson | first1=J. | title=Proof of a conjecture by Dyson in the statistical theory of energy levels | doi=10.1063/1.1724277 | mr=0148401 | year=1962 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=3 | pages=752–753 | issue=4}}
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| *{{Citation | last1=Macdonald | first1=I. G. | title=Some conjectures for root systems | doi=10.1137/0513070 | mr=674768 | year=1982 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=13 | issue=6 | pages=988–1007}}
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| *{{Citation | last1=Sills | first1=Andrew V. | title=Disturbing the Dyson conjecture, in a generally GOOD way | doi=10.1016/j.jcta.2005.12.005 | mr=2259066 | year=2006 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=113 | issue=7 | pages=1368–1380}}
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| *{{Citation | last1=Sills | first1=Andrew V. | last2=Zeilberger | first2=Doron | author2-link=Doron Zeilberger | title=Disturbing the Dyson conjecture (in a GOOD way) | url=http://projecteuclid.org/euclid.em/1175789739 | mr=2253005 | year=2006 | journal=Experimental Mathematics | issn=1058-6458 | volume=15 | issue=2 | pages=187–191 | doi=10.1080/10586458.2006.10128959}}
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| *{{Citation | last1=Wilson | first1=Kenneth G. | author1-link = Kenneth G. Wilson | title=Proof of a conjecture by Dyson | doi=10.1063/1.1724291 | mr=0144627 | year=1962 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=3 | pages=1040–1043 | issue=5}}
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| *{{Citation | last1=Zeilberger | first1=Doron | author1-link=Doron Zeilberger | last2=Bressoud | first2=David M. |author2-link=David Bressoud | title=A proof of Andrews' q-Dyson conjecture | doi=10.1016/0012-365X(85)90081-0 | mr=791661 | year=1985 | journal=[[Discrete Mathematics (journal)|Discrete Mathematics]] | issn=0012-365X | volume=54 | issue=2 | pages=201–224}}
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| [[Category:Enumerative combinatorics]]
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| [[Category:Algebraic combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Mathematical identities]]
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| [[Category:Conjectures]]
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| [[Category:Freeman Dyson]]
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I'm a 33 years old, married and study at the high school (Educational Policy Studies).
In my spare time I teach myself German. I've been twicethere and look forward to returning anytime soon. I love to read, preferably on my kindle. I really love to watch The Big Bang Theory and 2 Broke Girls as well as docus about anything geological. I like Stone collecting.
xunjie 国内の人々に影響を与える。
パーソナライズされた熟女との完全な衣装のシャツを表示す�
デザイナーのクリストフ·ルメールチップの材料にさらに注意を比較した。 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����ߥ�� �rӋ ���] バングラデシュの衣服の輸出は130億ドルのターゲットバングラデシュの4,000人以上の縫製工場が、
マグレディ分析ボーアファッション有限調印式のために演説を行った調印式の意義は成都テキスタイル·カレッジ·レディ·ボーア研究所だけでなく、
今シーズンの完全な焦点になりたい？今シーズンの販売マスターになりたい？ディWeinaはすぐ割引ブランドの仲間入りをし、 [http://www.tobler-verlag.ch/media/galerie/jp/shop/nb/ �˥�`�Х�� �����ȥ�å�] 次の確立ZigNanoで実行するための動きの異なるオプションを言及するとバスケットボール世界ZigEncore後、
その素朴な雰囲気のバッグ本体ラインと強力な機能を保持し、
チャンツィイーとSaが恋にBeiningエンターテインメントゴシップファン心配、[http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����ߥ�� �rӋ] 美しいレースが大きな金属バックルガードル著名なスタイル、
外の美しさの気質のバージョンは、
キャットアイサングラスの1組のシンプルな黒のパンツ、
財布やiPhoneケースを含む単一の特殊な製品の彼の英国風の創作を販売しています。 [http://belpars.by/Files/Config/tp/new/best/toms.html �ȥॺ ����
���å�]
my website; クロエ 財布