"Q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 27개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
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| − | + | ==q의 의미== | |
| − | + | * 양자를 뜻하는 quantum의 첫글자 | |
| + | * 극한 <math>q \to 1</math>로 갈 때, 고전적인 경우를 다시 얻게 된다 | ||
| + | * h를 파라메터로 사용하는 경우(플랑크상수에서 빌려옴), 극한 <math>h \to 0</math>를 통하여 고전적인 경우를 얻고, <math>q=e^h</math>를 만족시킨다 | ||
| + | * 유한체의 원소의 개수를 보통 q로 나타냄 | ||
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| − | + | ==실수의 q-analogue== | |
| − | * | + | * 실수 <math>\alpha</math>에 대하여 다음과 같이 정의:<math>[\alpha]_q =\frac{1-q^{\alpha}}{1-q} </math> |
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| − | + | * 극한 <math>q \to 1</math>:<math>\frac{1-q^{\alpha}}{1-q} \to \alpha</math> | |
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| − | + | ==q-차분연산자== | |
| − | * | + | * 미분에 대응:<math>D_qf(x)=\frac{f(x)-f(qx)}{x-qx}=\frac{f(x)-f(qx)}{(1-q)x}</math> |
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==basic 초기하급수 (q-초기하급수)== | ==basic 초기하급수 (q-초기하급수)== | ||
| − | * [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue | + | * [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue |
| − | * q-초기하급수 | + | :<math>_{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</math> |
| − | * | + | * q-초기하급수 또는 basic 초기하급수로 불림 |
| − | * [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]] | + | * 오일러의 [[자연수의 분할수(integer partitions)|분할수]]에 대한 연구에서 다음과 같은 등식이 얻어짐 :<math>\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)}</math> |
| − | + | * [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]] 을 이해하는 틀을 제공 | |
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| − | * <math> | + | ==q-초기하급수에 대한 오일러공식== |
| + | * [[오일러의 q-초기하급수에 대한 무한곱 공식]] | ||
| + | :<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> | ||
| + | * [[오일러의 오각수정리(pentagonal number theorem)]] | ||
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| − | + | ==q-초기하급수의 예== | |
| − | + | * [[q-이항정리]] | |
| + | :<math>\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</math> | ||
| + | ===로저스-라마누잔 항등식=== | ||
| + | :<math>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}</math> | ||
| + | :<math>H(q)=R(q)</math> | ||
| + | :<math>G(q)=R(1)</math> | ||
| + | * <math>j=k=0</math>, <math>z=-q^{\frac{1}{2}}</math> 인 경우 | ||
| + | :<math>G(q) =1+ \sum_{n=1}^\infty \frac {q^{n^2}} {(q;q)_n}</math> | ||
| − | < | + | * <math>j=k=0</math>, <math>z=-q^{\frac{3}{2}}</math> 인 경우:<math>H(q) =1+\sum_{n=1}^\infty \frac {q^{n^2+n}} {(q;q)_n}</math> |
| − | * | + | ===라마누잔의 mock 세타함수=== |
| + | * 라마누잔 3rd order mock 세타함수 | ||
| + | :<math>f(q)=1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}</math> | ||
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| − | + | ==삼중곱 공식== | |
| + | * [[자코비 삼중곱(Jacobi triple product)]] | ||
| + | :<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> | ||
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| − | + | ==메모== | |
| + | * Heine's theorem | ||
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| − | + | ==관련된 항목들== | |
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| − | + | * [[로저스-라마누잔 항등식]] | |
| + | * [[자코비 세타함수]] | ||
| + | * [[데데킨트 에타함수]] | ||
| + | * [[quantized universal enveloping algebra]] | ||
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| − | + | ==사전 형태의 자료== | |
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* 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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| − | + | ==리뷰, 에세이, 강의노트== | |
| − | * | + | * 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. |
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* 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 ( | + | * [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 |
| + | * Andrews, G. “Applications of Basic Hypergeometric Functions.” SIAM Review 16, no. 4 (October 1, 1974): 441–84. doi:[http://dx.doi.org/10.1137/1016081 10.1137/1016081.] | ||
| − | + | ==관련논문== | |
| − | + | * Hironori Mori, Takeshi Morita, Summation formulae for the bilateral basic hypergeometric series <math>{}_1ψ_1 ( a; b; q, z )</math>, http://arxiv.org/abs/1603.06657v1 | |
| − | + | * Ye, Runping, and Qing Zou. “On the Very-Well-Poised Bilateral Basic Hypergeometric <math>_5\psi_5</math> Series.” arXiv:1601.02074 [math], January 8, 2016. http://arxiv.org/abs/1601.02074. | |
| − | + | * [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities] | |
| − | < | + | ** McLaughlin, 2008 |
| − | + | * [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.274&rep=rep1&type=pdf The history of q-calculus and a new method] | |
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| − | * [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities] | ||
| − | ** McLaughlin, 2008 | ||
| − | * [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.274&rep=rep1&type=pdf The history of q-calculus and a new method] | ||
** T Ernst | ** T Ernst | ||
| − | * [http://www.sm.luth.se/%7Enorbert/home_journal/electronic/104art4.pdf A method for q-calculus] | + | * [http://www.sm.luth.se/%7Enorbert/home_journal/electronic/104art4.pdf A method for q-calculus] |
** T Ernst, Journal of Nonlinear Mathematical Physics, 2003 | ** T Ernst, Journal of Nonlinear Mathematical Physics, 2003 | ||
| − | * [http://arxiv.org/abs/math/9605230 Elementary derivations of summations and transformation formulas for q-series] | + | * [http://arxiv.org/abs/math/9605230 Elementary derivations of summations and transformation formulas for q-series] |
** George Gasper Jr, 1996 | ** George Gasper Jr, 1996 | ||
| − | + | ==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] | + | * [http://books.google.com/books?id=31l4uC7lqGAC&dq=Gasper,+George;+Rahman,+Mizan+%282004%29,+Basic+hypergeometric+series Basic hypergeometric series] |
** Gasper, George; Rahman, Mizan (2004) | ** Gasper, George; Rahman, Mizan (2004) | ||
| − | + | * [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus] | |
| − | + | ** 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] | |
| − | + | ** George E. Andrews, AMS Bookstore, 1986 | |
| − | + | * [http://www.amazon.com/B-Marko-Petkovsek/dp/1568810636 A=B] | |
| − | + | ** Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1 | |
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| − | * [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus] | ||
| − | ** Victor Kac, Pokman Cheung, | ||
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| − | * [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] | ||
| − | ** George E. Andrews, | ||
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| − | * [http://www.amazon.com/B-Marko-Petkovsek/dp/1568810636 A=B] | ||
| − | ** Marko Petkovsek, | ||
** [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] | ||
| − | + | [[분류:q-급수]] | |
| − | + | [[분류:특수함수]] | |
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| − | + | ==메타데이터== | |
| − | + | ===위키데이터=== | |
| − | * [ | + | * ID : [https://www.wikidata.org/wiki/Q1062958 Q1062958] |
| − | * [ | + | ===Spacy 패턴 목록=== |
| − | * [ | + | * [{'LOWER': 'basic'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}] |
| − | * | + | * [{'LOWER': 'q'}, {'OP': '*'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}] |
2021년 2월 17일 (수) 03:52 기준 최신판
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)
- 초기하급수의 q-analogue
\[_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right]=\sum_{n=0}^\infty \frac {(a_1;q)_n(a_2;q)_n\cdots (a_{j};q)_n} {(q;q)_n(b_1;q)_n,\cdots (b_k,q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n\]
- 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
\[\sum_{n=0}^{\infty} \frac{(a;q)_n}{(q;q)_n}z^n=\sum_{n=0}^{\infty} \frac{(1-a)^q_n}{(1-q)^q_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}=\prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}, |z|<1\]
로저스-라마누잔 항등식
\[R(z)=1+\sum_{n\geq 1}\frac{z^nq^{n^2}}{(1-q)\cdots(1-q^n)}=\sum_{n\geq 0}\frac{z^nq^{n^2}}{(1-q)_q^n}\] \[H(q)=R(q)\] \[G(q)=R(1)\]
- \(j=k=0\), \(z=-q^{\frac{1}{2}}\) 인 경우
\[G(q) =1+ \sum_{n=1}^\infty \frac {q^{n^2}} {(q;q)_n}\]
- \(j=k=0\), \(z=-q^{\frac{3}{2}}\) 인 경우\[H(q) =1+\sum_{n=1}^\infty \frac {q^{n^2+n}} {(q;q)_n}\]
라마누잔의 mock 세타함수
- 라마누잔 3rd order mock 세타함수
\[f(q)=1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\]
삼중곱 공식
\[\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\]
메모
- Heine's theorem
관련된 항목들
사전 형태의 자료
- http://en.wikipedia.org/wiki/Basic_hypergeometric_series
- http://en.wikipedia.org/wiki/Q-analog
- http://en.wikipedia.org/wiki/Q-derivative
- http://en.wikipedia.org/wiki/Quantum_calculus
리뷰, 에세이, 강의노트
- George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc. 44 (2007), 561-573.
- Koornwinder, Tom H. 1996. “Special functions and q-commuting variables”. q-alg/9608008 (8월 13). http://arxiv.org/abs/q-alg/9608008
- A brief introduction to the world of q , R Askey (in Symmetries and integrability of difference equations), 1996
- Andrews, G. “Applications of Basic Hypergeometric Functions.” SIAM Review 16, no. 4 (October 1, 1974): 441–84. doi:10.1137/1016081.
관련논문
- Hironori Mori, Takeshi Morita, Summation formulae for the bilateral basic hypergeometric series \({}_1ψ_1 ( a; b; q, z )\), http://arxiv.org/abs/1603.06657v1
- Ye, Runping, and Qing Zou. “On the Very-Well-Poised Bilateral Basic Hypergeometric \(_5\psi_5\) Series.” arXiv:1601.02074 [math], January 8, 2016. http://arxiv.org/abs/1601.02074.
- Rogers-Ramanujan-Slater Type identities
- McLaughlin, 2008
- The history of q-calculus and a new method
- T Ernst
- A method for q-calculus
- T Ernst, Journal of Nonlinear Mathematical Physics, 2003
- Elementary derivations of summations and transformation formulas for q-series
- George Gasper Jr, 1996
관련도서
- Basic hypergeometric series
- Gasper, George; Rahman, Mizan (2004)
- Quantum calculus
- Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
- q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra
- George E. Andrews, AMS Bookstore, 1986
- A=B
- Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1
- http://www.math.upenn.edu/~wilf/AeqB.html
메타데이터
위키데이터
- ID : Q1062958
Spacy 패턴 목록
- [{'LOWER': 'basic'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]
- [{'LOWER': 'q'}, {'OP': '*'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]