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=

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

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

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