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

수학노트
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잔글 (찾아 바꾸기 – “수학사연표” 문자열을 “수학사 연표” 문자열로)
 
(같은 사용자의 중간 판 15개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==이 항목의 스프링노트 원문주소==
 
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* [[q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|양자미적분학(q-calculus)]]<br>
 
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==개요==
 
==개요==
  
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==q의 의미==
 
==q의 의미==
  
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*  양자를 뜻하는 quantum의 첫글자<br>
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*  양자를 뜻하는 quantum의 첫글자
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*  극한 <math>q \to 1</math>로 갈 때, 고전적인 경우를 다시 얻게 된다<br>
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*  극한 <math>q \to 1</math>로 갈 때, 고전적인 경우를 다시 얻게 된다
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*  h를 파라메터로 사용하는 경우(플랑크상수에서 빌려옴), 극한 <math>h \to 0</math>를 통하여 고전적인 경우를 얻고, <math>q=e^h</math>를 만족시킨다<br>
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*  h를 파라메터로 사용하는 경우(플랑크상수에서 빌려옴), 극한 <math>h \to 0</math>를 통하여 고전적인 경우를 얻고, <math>q=e^h</math>를 만족시킨다
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*  유한체의 원소의 개수를 보통 q로 나타냄<br>
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*  유한체의 원소의 개수를 보통 q로 나타냄
  
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==실수의 q-analogue==
 
==실수의 q-analogue==
  
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*  실수 <math>\alpha</math>에 대하여 다음과 같이 정의:<math>[\alpha]_q =\frac{1-q^{\alpha}}{1-q} </math><br>
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*  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><br>
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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==
 
==q-차분연산자==
  
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*  미분에 대응:<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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*  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)==
  
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* [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue:<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>
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* [[초기하급수(Hypergeometric series)|초기하급수]]의 q-analogue
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* q-초기하급수 또는 basic 초기하급수로 불림
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:<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>
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*  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><br>
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* q-초기하급수 또는 basic 초기하급수로 불림
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]] 을 이해하는 틀을 제공<br>
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*  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>
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]] 을 이해하는 틀을 제공
  
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==q-초기하급수에 대한 오일러공식==
 
==q-초기하급수에 대한 오일러공식==
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* [[오일러의 q-초기하급수에 대한 무한곱 공식]]
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* 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>
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:<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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:<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>
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* [[오일러의 오각수정리(pentagonal number theorem)]]<br>
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* [[오일러의 오각수정리(pentagonal number theorem)]]
  
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==q-초기하급수의 예==
 
==q-초기하급수의 예==
  
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* [[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><br>
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* [[q-이항정리]]
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]의 중요한 예:<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><br>
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:<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>
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===로저스-라마누잔 항등식===
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:<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>
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:<math>H(q)=R(q)</math>
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:<math>G(q)=R(1)</math>
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* <math>j=k=0</math>, <math>z=-q^{\frac{1}{2}}</math> 인 경우
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:<math>G(q) =1+ \sum_{n=1}^\infty \frac {q^{n^2}} {(q;q)_n}</math>
  
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* <math>j=k=0</math>, <math>z=-q^{\frac{1}{2}}</math> 인 경우
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* <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>
  
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<math>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</math>
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===라마누잔의 mock 세타함수===
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* 라마누잔 3rd order mock 세타함수
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:<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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* <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} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots</math><br>
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==삼중곱 공식==
 
==삼중곱 공식==
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* [[자코비 삼중곱(Jacobi triple product)]]
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:<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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* [[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>
 
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==Heine's theorem==
 
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==역사==
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=Kummel+nome
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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* [[수학사 연표]]
 
  
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==메모==
 
==메모==
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* Heine's theorem
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==== 하위페이지 ====
 
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* [[q-초기하급수(q-hypergeometric series)와 양자미적분학(q-calculus)|양자미적분학(q-calculus)]]<br>
 
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** [[q-감마함수]]<br>
 
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** [[q-이항계수 (가우스 다항식)|q-이항계수(가우스 다항식)]]<br>
 
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*** [[q-이항계수의 목록]]<br>
 
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** [[q-이항정리]]<br>
 
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** [[q-적분 (잭슨 적분, Jackson integral)|q-적분]]<br>
 
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** [[q-팩토리얼]]<br>
 
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==관련된 항목들==
 
==관련된 항목들==
  
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* [[로저스-라마누잔 항등식|로저스-라마누잔 연분수와 항등식]]<br>
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* [[로저스-라마누잔 항등식]]
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* [[자코비 세타함수]]<br>
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* [[자코비 세타함수]]
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* [[데데킨트 에타함수]]<br>
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* [[데데킨트 에타함수]]
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* [[quantized universal enveloping algebra]]<br>
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* [[quantized universal enveloping algebra]]
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==수학용어번역==
 
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* http://www.google.com/dictionary?langpair=en|ko&q=
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=basic
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
  
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==사전 형태의 자료==
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==사전 형태의 자료==
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*   <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
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* 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
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* http://en.wikipedia.org/wiki/
 
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* http://www.wolframalpha.com/input/?i=
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
  
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==리뷰, 에세이, 강의노트==
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* '''[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.
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** 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
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* [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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* [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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* 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.]
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==관련논문==
 
==관련논문==
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* 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
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* [http://dx.doi.org/10.1023/A:1006949508631 Some Asymptotic Formulae for q-Shifted Factorials]<br>
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* 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.
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** Richard J. McIntosh, The Ramanujan Journal, 1999
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* [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities]
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* [http://jlms.oxfordjournals.org/cgi/content/short/51/1/120 Some Asymptotic Formulae for q-Hypergeometric Series]<br>
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**  McLaughlin, 2008
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** Richard J. McIntosh, Journal of the London Mathematical Society 1995 51(1):120-136
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* [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://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
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** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
 
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* [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities]<br>
 
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**  McLaughlin, 2008<br>
 
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* [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]<br>
 
 
** T Ernst
 
** T Ernst
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* [http://www.sm.luth.se/%7Enorbert/home_journal/electronic/104art4.pdf A method for q-calculus]<br>
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* [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
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* [http://arxiv.org/abs/math/9605230 Elementary derivations of summations and transformation formulas for q-series]<br>
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* [http://arxiv.org/abs/math/9605230 Elementary derivations of summations and transformation formulas for q-series]
 
** George Gasper Jr, 1996
 
** George Gasper Jr, 1996
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* [http://dx.doi.org/10.1137/1016081 Applications of Basic Hypergeometric Functions]<br>
 
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** George E. Andrews, SIAM Rev. Volume 16, Issue 4, pp. 441-484 (October 1974)
 
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://dx.doi.org/
 
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==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>
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* [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)
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* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]
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* [http://www.amazon.com/Quantum-Calculus-Victor-Kac/dp/0387953418 Quantum calculus]<br>
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** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
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** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
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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]
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** George E. Andrews, AMS Bookstore, 1986
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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]<br>
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* [http://www.amazon.com/B-Marko-Petkovsek/dp/1568810636 A=B]
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** George E. Andrews, AMS Bookstore, 1986
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** 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]<br>
 
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** Victor Kac, Pokman Cheung, Universitext, Springer-Verlag, 2002
 
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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]<br>
 
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** George E. Andrews, AMS Bookstore, 1986
 
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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>
 
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** Gasper, George; Rahman, Mizan (2004), 
 
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* [http://www.amazon.com/B-Marko-Petkovsek/dp/1568810636 A=B]<br>
 
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** Marko Petkovsek, Herbert Wilf and Doron Zeilberger, AK Peters, Ltd, 1996-1
 
 
** [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]
  
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*  도서내검색<br>
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[[분류:q-급수]]
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** http://books.google.com/books?q=
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[[분류:특수함수]]
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
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==블로그==
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1062958 Q1062958]
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===Spacy 패턴 목록===
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* [{'LOWER': 'basic'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]
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* [{'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)

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

  • 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



리뷰, 에세이, 강의노트

관련논문

관련도서

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'basic'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]
  • [{'LOWER': 'q'}, {'OP': '*'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'series'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Q-초기하급수(q-hypergeometric_series)와_양자미적분학(q-calculus)&oldid=52155"