"Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)"의 두 판 사이의 차이

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==q-초기하급수에 대한 오일러공식==
 
==q-초기하급수에 대한 오일러공식==
 
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* [[오일러의 q-초기하급수에 대한 무한곱 공식]]
* 오일러의 무한곱표현 '''[Andrews2007]'''
 
 
:<math>\prod_{n=0}^{\infty}(1+zq^n)=1+\sum_{n\geq 1}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math>
 
:<math>\prod_{n=0}^{\infty}(1+zq^n)=1+\sum_{n\geq 1}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math>
 
:<math>\prod_{n=0}^{\infty}\frac{1}{1-zq^n}=1+\sum_{n\geq 1}\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math><br>
 
:<math>\prod_{n=0}^{\infty}\frac{1}{1-zq^n}=1+\sum_{n\geq 1}\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math><br>
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==삼중곱 공식==
 
==삼중곱 공식==
 
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* [[자코비 삼중곱(Jacobi triple product)]]
* [[자코비 세타함수]]의 삼중곱 공식:<math>\sum_{n=-\infty}^\infty  z^{n}q^{n^2}= \prod_{m=1}^\infty  \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)</math><br>
+
:<math>\sum_{n=-\infty}^\infty  z^{n}q^{n^2}= \prod_{m=1}^\infty  \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)</math><br>
  
 
 
 
 
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==== 하위페이지 ====
 
 
* [[q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|양자미적분학(q-calculus)]]<br>
 
** [[q-감마함수]]<br>
 
** [[q-이항계수 (가우스 다항식)|q-이항계수(가우스 다항식)]]<br>
 
*** [[q-이항계수의 목록]]<br>
 
** [[q-이항정리]]<br>
 
** [[q-적분 (잭슨 적분, Jackson integral)|q-적분]]<br>
 
** [[q-팩토리얼]]<br>
 
 
 
 
 
 
 
 
 
 
  
 
==관련된 항목들==
 
==관련된 항목들==
  
* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]<br>
+
* [[로저스-라마누잔 항등식]]<br>
 
* [[자코비 세타함수]]<br>
 
* [[자코비 세타함수]]<br>
 
* [[데데킨트 에타함수]]<br>
 
* [[데데킨트 에타함수]]<br>
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==사전 형태의 자료==
 
==사전 형태의 자료==
 
*   <br>
 
 
* http://en.wikipedia.org/wiki/Basic_hypergeometric_series
 
* http://en.wikipedia.org/wiki/Basic_hypergeometric_series
 
* http://en.wikipedia.org/wiki/Q-analog
 
* http://en.wikipedia.org/wiki/Q-analog
 
* http://en.wikipedia.org/wiki/Q-derivative
 
* http://en.wikipedia.org/wiki/Q-derivative
* http://en.wikipedia.org/wiki/Quantum_calculus[http://en.wikipedia.org/wiki/Quantum_calculus ]
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* http://en.wikipedia.org/wiki/Quantum_calculus
* http://en.wikipedia.org/wiki/
 
* http://www.wolframalpha.com/input/?i=
 
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
** http://www.research.att.com/~njas/sequences/?q=
 
  
 
 
  
 
 
 
 
  
 
 
 
 
 
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==리뷰논문, 에세이, 강의노트==
* '''[Andrews2007]'''[http://www.ams.org/bull/2007-44-04/S0273-0979-07-01180-9/ Euler's "De Partitio Numerorum"]<br>
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* George E. Andrews[http://www.ams.org/bull/2007-44-04/S0273-0979-07-01180-9/ Euler's "De Partitio Numerorum"], Bull. Amer. Math. Soc. 44 (2007), 561-573.
** George E. Andrews, Bull. Amer. Math. Soc. 44 (2007), 561-573.
 
 
* Koornwinder, Tom H. 1996. “Special functions and q-commuting variables”. <em>q-alg/9608008</em> (8월 13). http://arxiv.org/abs/q-alg/9608008
 
* Koornwinder, Tom H. 1996. “Special functions and q-commuting variables”. <em>q-alg/9608008</em> (8월 13). http://arxiv.org/abs/q-alg/9608008
 
* [http://books.google.com/books?id=RuUbBajmhgwC&pg=PA13&hl=ko&source=gbs_toc_r&cad=9#v=onepage&q=&f=false A brief introduction to the world of q] , R Askey (in Symmetries and integrability of difference equations), 1996
 
* [http://books.google.com/books?id=RuUbBajmhgwC&pg=PA13&hl=ko&source=gbs_toc_r&cad=9#v=onepage&q=&f=false A brief introduction to the world of q] , R Askey (in Symmetries and integrability of difference equations), 1996
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==관련논문==
 
==관련논문==
 
* [http://dx.doi.org/10.1023/A:1006949508631 Some Asymptotic Formulae for q-Shifted Factorials]<br>
 
** Richard J. McIntosh, The Ramanujan Journal, 1999
 
* [http://jlms.oxfordjournals.org/cgi/content/short/51/1/120 Some Asymptotic Formulae for q-Hypergeometric Series]<br>
 
** Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
 
 
* [http://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
* [http://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
 
** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
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* [http://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
* [http://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
 
** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://dx.doi.org/
 
 
 
 
  
 
 
  
 
==관련도서==
 
==관련도서==
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* [http://books.google.com/books?id=31l4uC7lqGAC&dq=Gasper,+George;+Rahman,+Mizan+%282004%29,+Basic+hypergeometric+series Basic hypergeometric series]<br>
 
* [http://books.google.com/books?id=31l4uC7lqGAC&dq=Gasper,+George;+Rahman,+Mizan+%282004%29,+Basic+hypergeometric+series Basic hypergeometric series]<br>
 
** Gasper, George; Rahman, Mizan (2004)
 
** Gasper, George; Rahman, Mizan (2004)
 
 
* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]<br>
 
* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]<br>
 
** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
 
** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
 
 
* [http://books.google.com/books?id=6aPO5PUh4qAC&dq=q-series:+their+development+and+application+in+analysis,+number+theory,+combinatorics,+physics,+and+computer+algebra&printsec=frontcover&source=bl&ots=s16bchosJl&sig=v3Dzc5hQScB6_TSQljbspGsa-xM&hl=ko&ei=B6CgSsTLE478sgOyhoGNDw&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra]<br>
 
* [http://books.google.com/books?id=6aPO5PUh4qAC&dq=q-series:+their+development+and+application+in+analysis,+number+theory,+combinatorics,+physics,+and+computer+algebra&printsec=frontcover&source=bl&ots=s16bchosJl&sig=v3Dzc5hQScB6_TSQljbspGsa-xM&hl=ko&ei=B6CgSsTLE478sgOyhoGNDw&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra]<br>
 
** George E. Andrews, AMS Bookstore, 1986
 
** George E. Andrews, AMS Bookstore, 1986
 
 
* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]<br>
 
* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]<br>
 
** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
 
** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
 
 
* [http://books.google.com/books?id=6aPO5PUh4qAC&dq=q-series:+their+development+and+application+in+analysis,+number+theory,+combinatorics,+physics,+and+computer+algebra&printsec=frontcover&source=bl&ots=s16bchosJl&sig=v3Dzc5hQScB6_TSQljbspGsa-xM&hl=ko&ei=B6CgSsTLE478sgOyhoGNDw&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra]<br>
 
* [http://books.google.com/books?id=6aPO5PUh4qAC&dq=q-series:+their+development+and+application+in+analysis,+number+theory,+combinatorics,+physics,+and+computer+algebra&printsec=frontcover&source=bl&ots=s16bchosJl&sig=v3Dzc5hQScB6_TSQljbspGsa-xM&hl=ko&ei=B6CgSsTLE478sgOyhoGNDw&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra]<br>
 
** George E. Andrews, AMS Bookstore, 1986
 
** George E. Andrews, AMS Bookstore, 1986
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** [http://www.math.upenn.edu/%7Ewilf/AeqB.html http://www.math.upenn.edu/~wilf/AeqB.html]
 
** [http://www.math.upenn.edu/%7Ewilf/AeqB.html http://www.math.upenn.edu/~wilf/AeqB.html]
  
*  도서내검색<br>
 
** http://books.google.com/books?q=
 
** http://book.daum.net/search/contentSearch.do?query=
 
*  도서검색<br>
 
** http://books.google.com/books?q=
 
** http://book.daum.net/search/mainSearch.do?query=
 
** http://book.daum.net/search/mainSearch.do?query=
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
==블로그==
 
 
[[분류:q-급수]]
 
[[분류:q-급수]]

2013년 3월 14일 (목) 10:10 판

개요

 

 

 

q의 의미

  • 양자를 뜻하는 quantum의 첫글자
  • 극한 \(q \to 1\)로 갈 때, 고전적인 경우를 다시 얻게 된다
  • h를 파라메터로 사용하는 경우(플랑크상수에서 빌려옴), 극한 \(h \to 0\)를 통하여 고전적인 경우를 얻고, \(q=e^h\)를 만족시킨다
  • 유한체의 원소의 개수를 보통 q로 나타냄

 

 

실수의 q-analogue

  • 실수 \(\alpha\)에 대하여 다음과 같이 정의\[[\alpha]_q =\frac{1-q^{\alpha}}{1-q} \]
  • 극한 \(q \to 1\)\[\frac{1-q^{\alpha}}{1-q} \to \alpha\]

 

 

q-차분연산자

  • 미분에 대응\[D_qf(x)=\frac{f(x)-f(qx)}{x-qx}=\frac{f(x)-f(qx)}{(1-q)x}\]

 

 

basic 초기하급수 (q-초기하급수)

\[_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right]=\sum_{n=0}^\infty \frac {(a_1;q)_n(a_2;q)_n\cdots (a_{j};q)_n} {(q;q)_n(b_1;q)_n,\cdots (b_k,q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n\]

  • q-초기하급수 또는 basic 초기하급수로 불림
  • 오일러의 분할수에 대한 연구에서 다음과 같은 등식이 얻어짐 \[\sum_{n=0}^\infty p(n)q^n = \prod_{n=1}^\infty \frac {1}{1-q^n} = \prod_{n=1}^\infty (1-q^n)^{-1} =1+\sum_{n=1}\frac{q^n}{(1-q)(1-q^2)\cdots(1-q^n)}\]
  • 로저스-라마누잔 연분수와 항등식 을 이해하는 틀을 제공

 

 

 

q-초기하급수에 대한 오일러공식

\[\prod_{n=0}^{\infty}(1+zq^n)=1+\sum_{n\geq 1}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\] \[\prod_{n=0}^{\infty}\frac{1}{1-zq^n}=1+\sum_{n\geq 1}\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\]

 

 

q-초기하급수의 예

  • q-이항정리\[\sum_{n=0}^{\infty} \frac{(a;q)_n}{(q;q)_n}z^n=\sum_{n=0}^{\infty} \frac{(1-a)^q_n}{(1-q)^q_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}=\prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}, |z|<1\]
  • 로저스-라마누잔 연분수와 항등식의 중요한 예\[R(z)=1+\sum_{n\geq 1}\frac{z^nq^{n^2}}{(1-q)\cdots(1-q^n)}=\sum_{n\geq 0}\frac{z^nq^{n^2}}{(1-q)_q^n}\]\[H(q)=R(q)\]\[G(q)=R(1)\]
  • \(j=k=0\), \(z=-q^{\frac{1}{2}}\) 인 경우

\[G(q) =1+ \sum_{n=1}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots\]

  • \(j=k=0\), \(z=-q^{\frac{3}{2}}\) 인 경우\[H(q) =1+\sum_{n=1}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\]

 

 

삼중곱 공식

\[\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\]

 

 

Heine's theorem

 

 

 

역사

 

 

메모

 


관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료


 

 

리뷰논문, 에세이, 강의노트

 

 

관련논문


관련도서