"Ramanujan's Cubic Continued fractions"의 두 판 사이의 차이
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| 6번째 줄: | 6번째 줄: | ||
<math>\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}} </math> | <math>\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}} </math> | ||
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| + | <math>\frac{\Gamma(\frac{1}{6})\Gamma(\frac{5}{6})}{\Gamma(\frac{3}{6})^{3}}=2</math> | ||
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| 13번째 줄: | 17번째 줄: | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | + | <math>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</math> | |
| 60번째 줄: | 64번째 줄: | ||
* [http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7343.pdf On Ramanujan’s cubic continued fraction]<br> | * [http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7343.pdf On Ramanujan’s cubic continued fraction]<br> | ||
** Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995)<br> | ** Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995)<br> | ||
| + | * <br>[http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F3%2F231&ei=JY1hSLWRLpSY8gSI7JSiBQ&usg=AFQjCNElhd9FwCl3m3Qcb3hW7j87K1P5FQ&sig2=4OhMIB56amm8h4EOGNSk6g Theorems Stated by Ramanujan (IX): Two Continued Fractions.]<br>[http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F3%2F231&ei=JY1hSLWRLpSY8gSI7JSiBQ&usg=AFQjCNElhd9FwCl3m3Qcb3hW7j87K1P5FQ&sig2=4OhMIB56amm8h4EOGNSk6g ][http://www.google.com/url?sa=t&ct=res&cd=1&url=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Freprint%2Fs1-4%2F3%2F231&ei=JY1hSLWRLpSY8gSI7JSiBQ&usg=AFQjCNElhd9FwCl3m3Qcb3hW7j87K1P5FQ&sig2=4OhMIB56amm8h4EOGNSk6g ]1929<br> | ||
http://www.emis.de/journals/ETNA/vol.25.2006/pp158-165.dir/pp158-165.pdf | http://www.emis.de/journals/ETNA/vol.25.2006/pp158-165.dir/pp158-165.pdf | ||
2010년 7월 22일 (목) 14:52 판
introduction
\({1 \over 1+} {q+q^2 \over 1+} {q^{2}+a^{4} \over 1+} {q^3+q^6 \over 1+}{\cdots} =\frac{(q;q^{2})_{\infty}}{(q^{3};q^{6})^{3}_{\infty}}\)
\(\frac{q^{1/3}}{1} {\ \atop+} \frac{q+q^2}{1}{\ \atop+} \frac{q^2+q^4}{1} {\ \atop+\dots}=q^{1/3}\frac{(q;q^{2})_{\infty}}{(q^{3};q^{6})^{3}_{\infty}} \)
\(\frac{\Gamma(\frac{1}{6})\Gamma(\frac{5}{6})}{\Gamma(\frac{3}{6})^{3}}=2\)
history
\(G(q)= \frac{q^{1/3}}{1} {\ \atop+} \frac{q+q^2}{1}{\ \atop+} \frac{q^2+q^4}{1} {\ \atop+\dots} \quad |q|<1\)
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[[4909919|]]
articles
- A new proof of two identities involving Ramanujan’s cubic continued fraction
- Chan, H.-C, 2010
- 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
- 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)
- 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
- Chan, H.-C, Int. J. Number Theory
- On Ramanujan’s cubic continued fraction
- Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995)
- Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995)
-
Theorems Stated by Ramanujan (IX): Two Continued Fractions.
[1][2]1929
http://www.emis.de/journals/ETNA/vol.25.2006/pp158-165.dir/pp158-165.pdf
Ramanujan's class invariants and cubic continued fraction
Berndt, 1995
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