"Ramanujan's Cubic Continued fractions"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| − | + | ==introduction== | |
| − | <math>{1 \over 1+} {q+q^2 \over 1+} {q^4 \over 1+} {q^3+q^6 \over 1+}{ | + | <math>{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}}</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> | |
| − | < | + | <math>\frac{\Gamma(\frac{1}{6})\Gamma(\frac{3}{6})\Gamma(\frac{5}{6})}{\Gamma(\frac{3}{6})^{3}}=2</math> |
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| − | + | ==history== | |
| − | + | * 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> | |
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| − | + | ==articles== | |
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| − | + | * A new proof of two identities involving Ramanujan’s cubic continued fraction | |
| + | ** Chan, H.-C, 2010 | ||
| + | * [http://arxiv.org/abs/math/0502323 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 | ||
| + | * [http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7343.pdf On Ramanujan’s cubic continued fraction] | ||
| + | ** Heng Huat Chan, ACTA ARITHMETICA. LXXIII.4 (1995) | ||
| + | * [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.] | ||
| + | ** Watson, G. N. 1929 | ||
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| − | + | Ramanujan's class invariants and cubic continued fraction | |
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| − | + | Berndt, 1995 | |
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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 | ||
| − | + | [http://www.emis.de/journals/ETNA/vol.25.2006/pp158-165.dir/pp158-165.pdf ] | |
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| − | + | [[분류:개인노트]] | |
| − | + | [[분류:q-series]] | |
| − | + | [[분류:migrate]] | |
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2020년 12월 28일 (월) 05:03 기준 최신판
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
\(G(q)= \frac{q^{1/3}}{1} {\ \atop+} \frac{q+q^2}{1}{\ \atop+} \frac{q^2+q^4}{1} {\ \atop+\dots} \quad |q|<1\)
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