Ramanujan's Cubic Continued fractions

introduction

\({1 \over 1+} {q+q^2 \over 1+} {q^{2}+a^{4} \over 1+} {q^3+q^6 \over 1+}{\cdots} =\frac{(q;q^{2})_{\infty}}{(q^{3};q^{6})^{3}_{\infty}}\)

\(\frac{q^{1/3}}{1} {\ \atop+} \frac{q+q^2}{1}{\ \atop+} \frac{q^2+q^4}{1} {\ \atop+\dots}=q^{1/3}\frac{(q;q^{2})_{\infty}}{(q^{3};q^{6})^{3}_{\infty}} \)

\(\frac{\Gamma(\frac{1}{6})\Gamma(\frac{3}{6})\Gamma(\frac{5}{6})}{\Gamma(\frac{3}{6})^{3}}=2\)

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

\(G(q)= \frac{q^{1/3}}{1} {\ \atop+} \frac{q+q^2}{1}{\ \atop+} \frac{q^2+q^4}{1} {\ \atop+\dots} \quad |q|<1\)

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

• A new proof of two identities involving Ramanujan’s cubic continued fraction
  
  • Chan, H.-C, 2010
    
• On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
  
  • C. Adiga, T. Kim, M.S.Mahadeva Naika, H. S. Madhusudhan, 2005
    
• Some evaluations of Ramanujan’s cubic continued fraction(http://www.zentralblatt-math.org/zmath/search/?an=1148.11303)
  
  • Bhargava, S., Vasuki, K.R., Sreeramamurthy, T.G., Indian J. Pure Appl. Math. 35, 1003–1025 (2004)
    
• Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function.
  
  • Chan, H.-C,  Int. J. Number Theory
    
• On Ramanujan’s cubic continued fraction
  
  • Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995)
    
• Theorems Stated by Ramanujan (IX): Two Continued Fractions.
  
  • Watson, G. N. 1929
    

 

Ramanujan's class invariants and cubic continued fraction

Berndt, 1995

• http://www.ams.org/mathscinet
• [1]http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/

 

http://www.emis.de/journals/ETNA/vol.25.2006/pp158-165.dir/pp158-165.pdf

[2]

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/