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이 항목의 스프링노트 원문주소==
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소== |
* [[q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|양자미적분학(q-calculus)]]<br> | * [[q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|양자미적분학(q-calculus)]]<br> | ||
| 7번째 줄: | 7번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요== |
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| − | <h5 style="margin: 0px; line-height: 2em;">q의 의미 | + | <h5 style="margin: 0px; line-height: 2em;">q의 의미== |
* 양자를 뜻하는 quantum의 첫글자<br> | * 양자를 뜻하는 quantum의 첫글자<br> | ||
| 26번째 줄: | 26번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">실수의 q-analogue | + | <h5 style="margin: 0px; line-height: 2em;">실수의 q-analogue== |
* 실수 <math>\alpha</math>에 대하여 다음과 같이 정의<br><math>[\alpha]_q =\frac{1-q^{\alpha}}{1-q} </math><br> | * 실수 <math>\alpha</math>에 대하여 다음과 같이 정의<br><math>[\alpha]_q =\frac{1-q^{\alpha}}{1-q} </math><br> | ||
| 36번째 줄: | 36번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">q-차분연산자 | + | <h5 style="margin: 0px; line-height: 2em;">q-차분연산자== |
* 미분에 대응<br><math>D_qf(x)=\frac{f(x)-f(qx)}{x-qx}=\frac{f(x)-f(qx)}{(1-q)x}</math><br> | * 미분에 대응<br><math>D_qf(x)=\frac{f(x)-f(qx)}{x-qx}=\frac{f(x)-f(qx)}{(1-q)x}</math><br> | ||
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| − | ==basic 초기하급수 (q-초기하급수) | + | ==basic 초기하급수 (q-초기하급수)== |
* [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue<br><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]</math> <math>=\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><br> | * [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue<br><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]</math> <math>=\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><br> | ||
| 57번째 줄: | 57번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">q-초기하급수에 대한 오일러공식 | + | <h5 style="margin: 0px; line-height: 2em;">q-초기하급수에 대한 오일러공식== |
* 오일러의 무한곱표현 '''[Andrews2007]'''<br><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><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> | * 오일러의 무한곱표현 '''[Andrews2007]'''<br><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><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> | ||
| 66번째 줄: | 66번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">q-초기하급수의 예 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">q-초기하급수의 예== |
* [[q-이항정리]]<br><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><br> | * [[q-이항정리]]<br><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><br> | ||
| 81번째 줄: | 81번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">삼중곱 공식 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">삼중곱 공식== |
* [[자코비 세타함수]]의 삼중곱 공식<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> | * [[자코비 세타함수]]의 삼중곱 공식<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> | ||
| 89번째 줄: | 89번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Heine's theorem | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">Heine's theorem== |
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사== |
* http://www.google.com/search?hl=en&tbs=tl:1&q=Kummel+nome | * http://www.google.com/search?hl=en&tbs=tl:1&q=Kummel+nome | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">메모 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">메모== |
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 항목들 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 항목들== |
* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]<br> | * [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]<br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역== |
* http://www.google.com/dictionary?langpair=en|ko&q= | * http://www.google.com/dictionary?langpair=en|ko&q= | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사전 형태의 자료 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사전 형태의 자료== |
* <br> | * <br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문== |
* [http://dx.doi.org/10.1023/A:1006949508631 Some Asymptotic Formulae for q-Shifted Factorials]<br> | * [http://dx.doi.org/10.1023/A:1006949508631 Some Asymptotic Formulae for q-Shifted Factorials]<br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서== |
* [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> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사== |
* 네이버 뉴스 검색 (키워드 수정)<br> | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
2012년 11월 1일 (목) 10:23 판
이 항목의 스프링노트 원문주소==
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
\(\sum_{n=0}^\infty p(n)q^n = \prod_{n=1}^\infty \frac {1}{1-q^n} \right = \prod_{n=1}^\infty (1-q^n)^{-1} =1+\sum_{n=1}\frac{q^n}{(1-q)(1-q^2)\cdots(1-q^n)}\)
- 로저스-라마누잔 연분수와 항등식 을 이해하는 틀을 제공
q-초기하급수에 대한 오일러공식==
- 오일러의 무한곱표현 [Andrews2007]
\(\prod_{n=0}^{\infty}(1+zq^n)=1+\sum_{n\geq 1}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
\(\prod_{n=0}^{\infty}\frac{1}{1-zq^n}=1+\sum_{n\geq 1}\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
- 오일러의 오각수정리(pentagonal number theorem)
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==
- 자코비 세타함수의 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==
하위페이지
관련된 항목들==
수학용어번역==
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[1]
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
- [Andrews2007]Euler's "De Partitio Numerorum"
- George E. Andrews, 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
관련논문==
- Some Asymptotic Formulae for q-Shifted Factorials
- Richard J. McIntosh, The Ramanujan Journal, 1999
- Some Asymptotic Formulae for q-Hypergeometric Series
- Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
- Applications of Basic Hypergeometric Functions
- George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
- 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
- Applications of Basic Hypergeometric Functions
- 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==
- 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
- 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
- Basic hypergeometric series
- Gasper, George; Rahman, Mizan (2004),
- A=B
- Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1
- http://www.math.upenn.edu/~wilf/AeqB.html
관련기사==
블로그==
- 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
- 실수 \(\alpha\)에 대하여 다음과 같이 정의
\([\alpha]_q =\frac{1-q^{\alpha}}{1-q} \)
- 극한 \(q \to 1\)
\(\frac{1-q^{\alpha}}{1-q} \to \alpha\)
- 미분에 대응
\(D_qf(x)=\frac{f(x)-f(qx)}{x-qx}=\frac{f(x)-f(qx)}{(1-q)x}\)
\(_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right]\) \(=\sum_{n=0}^\infty \frac {(a_1;q)_n(a_2;q)_n\cdots (a_{j};q)_n} {(q;q)_n(b_1;q)_n,\cdots (b_k,q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n\)
\(\sum_{n=0}^\infty p(n)q^n = \prod_{n=1}^\infty \frac {1}{1-q^n} \right = \prod_{n=1}^\infty (1-q^n)^{-1} =1+\sum_{n=1}\frac{q^n}{(1-q)(1-q^2)\cdots(1-q^n)}\)
- 오일러의 무한곱표현 [Andrews2007]
\(\prod_{n=0}^{\infty}(1+zq^n)=1+\sum_{n\geq 1}\frac{q^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
\(\prod_{n=0}^{\infty}\frac{1}{1-zq^n}=1+\sum_{n\geq 1}\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\) - 오일러의 오각수정리(pentagonal number theorem)
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
\(H(q) =1+\sum_{n=1}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\)
\(\sum_{n=0}^{\infty} \frac{(a;q)_n}{(q;q)_n}z^n=\sum_{n=0}^{\infty} \frac{(1-a)^q_n}{(1-q)^q_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}=\prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}, |z|<1\)
\(R(z)=1+\sum_{n\geq 1}\frac{z^nq^{n^2}}{(1-q)\cdots(1-q^n)}=\sum_{n\geq 0}\frac{z^nq^{n^2}}{(1-q)_q^n}\)
\(H(q)=R(q)\)
\(G(q)=R(1)\)
\(H(q) =1+\sum_{n=1}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots\)
삼중곱 공식==
- 자코비 세타함수의 삼중곱 공식
\(\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\)
\(\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\)
Heine's theorem==
역사==
메모==
하위페이지
관련된 항목들==
수학용어번역==
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[1]
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
- [Andrews2007]Euler's "De Partitio Numerorum"
- George E. Andrews, 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
- George E. Andrews, Bull. Amer. Math. Soc. 44 (2007), 561-573.
관련논문==
- Some Asymptotic Formulae for q-Shifted Factorials
- Richard J. McIntosh, The Ramanujan Journal, 1999
- Some Asymptotic Formulae for q-Hypergeometric Series
- Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
- Applications of Basic Hypergeometric Functions
- George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
- 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
- Applications of Basic Hypergeometric Functions
- 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/
- Richard J. McIntosh, The Ramanujan Journal, 1999
- Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
- George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
- McLaughlin, 2008
- T Ernst
- T Ernst, Journal of Nonlinear Mathematical Physics, 2003
- George Gasper Jr, 1996
- George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
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
- 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
- Basic hypergeometric series
- Gasper, George; Rahman, Mizan (2004),
- A=B
- Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1
- http://www.math.upenn.edu/~wilf/AeqB.html
- Gasper, George; Rahman, Mizan (2004)
- Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
- George E. Andrews, AMS Bookstore, 1986
- Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
- George E. Andrews, AMS Bookstore, 1986
- Gasper, George; Rahman, Mizan (2004),
- Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1
- http://www.math.upenn.edu/~wilf/AeqB.html