Explicit evaluations of a Ramanujan-Selberg continued fraction

Zhang, Liang-Cheng

doi:10.1090/S0002-9939-01-06183-4

Explicit evaluations of a Ramanujan-Selberg continued fraction
HTML articles powered by AMS MathViewer

by Liang-Cheng Zhang PDF

Proc. Amer. Math. Soc. 130 (2002), 9-14 Request permission

Abstract:

This paper gives explicit evaluations for a Ramanujan-Selberg continued fraction in terms of class invariants and singular moduli.

References

G. E. Andrews, Bruce C. Berndt, Lisa Jacobsen, and Robert L. Lamphere, The continued fractions found in the unorganized portions of Ramanujan’s notebooks, Mem. Amer. Math. Soc. 99 (1992), no. 477, vi+71. MR 1124109, DOI 10.1090/memo/0477
Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903, DOI 10.1007/978-1-4612-0965-2
Bruce C. Berndt, Ramanujan’s notebooks. Part V, Springer-Verlag, New York, 1998. MR 1486573, DOI 10.1007/978-1-4612-1624-7
Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, J. Reine Angew. Math. 480 (1996), 141–159. MR 1420561, DOI 10.1515/crll.1996.480.141
Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ramanujan’s singular moduli, Ramanujan J. 1 (1997), no. 1, 53–74. MR 1607528, DOI 10.1023/A:1009715221401
Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2125–2173. MR 1376539, DOI 10.1090/S0002-9947-97-01738-8
B. J. Birch, Weber’s class invariants, Mathematika 16 (1969), 283–294. MR 262206, DOI 10.1112/S0025579300008251
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. MR 877728
David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
E. Heine, Untersuchungen über die Reihe $1 + \frac { (1-q^\alpha ) (1-q^\beta ) } { (1-q) (1-q^\gamma ) } x + \frac { (1-q^\alpha ) (1-q^{\alpha +1}) (1-q^\beta ) (1-q^{\beta +1})} { (1-q)(1-q^2) (1-q^\gamma ) (1-q^{\gamma +1})} x^{2} + \dots$, J. Reine Angew. Math. 34 (1847), 285-328.
Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960, DOI 10.1007/978-1-4612-4752-4
K. G. Ramanathan, Ramanujan’s continued fraction, Indian J. Pure Appl. Math. 16 (1985), no. 7, 695–724. MR 801801
K. G. Ramanathan, Some applications of Kronecker’s limit formula, J. Indian Math. Soc. (N.S.) 52 (1987), 71–89 (1988). MR 989231
Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
S. Ramanujan, Modular equations and approximations to $\pi$, Quart. J. Math. 3 (1914), 81-98.
Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735
S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
A. Selberg, Über einige arithmetische Identitäten, Avh. Norske Vid.-Akad. Oslo I, Mat.-Natur. Kl., (1936), 2-23.
G. N. Watson, Theorems stated by Ramanujan (VII): theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
H. Weber, Lehrbuch der Algebra, dritter Band, Chelsea, New York, 1961.
Liang Cheng Zhang, $q$-difference equations and Ramanujan-Selberg continued fractions, Acta Arith. 57 (1991), no. 4, 307–355. MR 1109991, DOI 10.4064/aa-57-4-307-355
Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula and modular equations. II, Analytic number theory, Vol. 2 (Allerton Park, IL, 1995) Progr. Math., vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 817–838. MR 1409396
Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula and modular equations. III, Acta Arith. 82 (1997), no. 4, 379–392. MR 1483690, DOI 10.4064/aa-82-4-379-392