Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

From a Ramanujan-Selberg continued fraction to a Jacobian identity

Author(s): Hei-Chi Chan
Journal: Proc. Amer. Math. Soc. 137 (2009), 2849-2856.
MSC (2000): Primary 05A15, 05A30, 05A40
Posted: March 4, 2009
MathSciNet review: 2506441
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Jacobi proved an elegant identity involving eight-fold infinite products. In this paper, we give a new proof of this identity. A key ingredient of our proof is an identity satisfied by a Ramanujan-Selberg continued fraction.

References:

1.: G. E. Andrews, On q-difference equations for certain well-poised basic hypergeometric series, Quart. J. Math. Oxford Ser. (2) 19 (1968), 433-447. MR 0237831 (38:6112)
2.: G. E. Andrews, The theory of partitions, Encycl. Math. and its Appl., vol. 2, G.-C. Rota, ed., Addison-Wesley, Reading, MA, 1976 (Reissued: Cambridge University Press, Cambridge, 1998). MR 0557013 (58:27738)
3.: G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
4.: G. E. Andrews, B. C. Berndt, L. Jacobsen and R. L. Lamphere, The continued fractions found in the unorganized portions of Ramanujan's notebooks, Mem. Amer. Math. Soc. 99 (1992), no. 477. MR 1124109 (93f:11008)
5.: G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer-Verlag, New York, 2005. MR 2135178 (2005m:11001)
6.: G. E. Andrews and K. Eriksson, Integer partitions, Cambridge University Press, Cambridge, 2004. MR 2122332 (2006b:11125)
7.: N. D. Baruah and N. Saikia, Modular relations and explicit values of Ramanujan-Selberg continued fractions, Int. J. Math. Math. Sci. 2006, Art. ID 54901, 1-15. MR 2251712 (2007e:11148)
8.: B. C. Berndt, Ramanujan's notebooks, Part III, Springer-Verlag, New York, 1991. MR 1117903 (92j:01069)
9.: B. C. Berndt, Number theory in the spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006. MR 2246314 (2007f:11001)
10.: B. C. Berndt, H. H. Chan, S.-S. Huang, S.-Y. Kang, J. Sohn, and S. H. Son, The Rogers-Ramanujan continued fraction, J. Comput. Appl. Math. 105 (1999), 9-24. MR 1690576 (2000b:11009)
11.: J. M. Borwein and P. B. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, Wiley, New York, 1987. MR 877728 (89a:11134)
12.: H. C. Chan, Ramanujan's cubic continued fraction and an analog of his ``most beautiful identity'', to appear in International Journal of Number Theory.
13.: H. C. Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, to appear in International Journal of Number Theory.
14.: H. C. Chan, Distribution of a certain partition function modulo powers of primes, submitted.
15.: H. C. Chan, A new proof for two identities involving the Ramanujan's cubic continued fraction, submitted.
16.: H. C. Chan and S. Ebbing, Fractorization theorems for the Rogers-Ramanujan continued fraction in the lost notebook, submitted.
17.: H. H. Chan and S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997), 75-90. MR 1607529 (99k:11015)
18.: W. Chu and L. Di Claudio, Classical Partition Identities and Basic Hypergeometric Series, Università degli Studi di Lecce, Lecce, Italy, 2004.
19.: J. A. Ewell, A note on a Jacobian identity, Proc. Amer. Math. Soc. 126 (1998), 421-423. MR 1451797 (98k:33030)
20.: H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154-190. MR 0211973 (35:2848)
21.: B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741-748. MR 0184001 (32:1477)
22.: M. D. Hirschhorn, An identity of Ramanujan, and applications, in q-Series from a Contemporary Perspective, Contemporary Mathematics, vol. 254, American Mathematical Society, Providence, RI, 2000, 229-234. MR 1768930 (2001f:11180)
23.: M. D. Hirschhorn, Ramanujan's ``most beautiful identity'', Austral. Math. Soc. Gazette 32 (2005), 259-262. MR 2176248 (2006e:11158)
24.: K. G. Ramanathan, Ramanujan's continued fraction, Indian J. Pure Appl. Math. 16 (1985), 695-724. MR 801801 (87e:11015)
25.: S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
26.: A. Selberg, Über einige arithmetische Identitäten, Avh. Norske Vid.-Akad. Oslo I. Mat.-Naturv. Kl. 8 (1936), 3-23.
27.: G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
28.: G. N. Watson, Theorems stated by Ramanujan (IX): Two continued fractions, J. London Math. Soc. 4 (1929), 231-237.
29.: E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, New York, 1973. MR 1424469 (97k:01072)
30.: L.-C. Zhang, q-difference equations and Ramanujan-Selberg continued fractions, Acta Arith. 57 (1991), 307-355. MR 1109991 (92j:11008)
31.: L.-C. Zhang, Ramanujan's class invariants, Kronecker's limit formula and modular equations. II, Analytic Number Theory, Proceedings of a Conference in Honor of H. Halberstam, vol. 2, Birkhäuser, Boston, MA, 1996, pp. 817-838. MR 1409396 (97i:11040)
32.: L.-C. Zhang, Ramanujan's class invariants, Kronecker's limit formula and modular equations. III, Acta Arith. LXXXII.4 (1997), 379-392. MR 1483690 (98k:11045)
33.: L.-C. Zhang, Explicit evaluations of a Ramanujan-Selberg continued fraction, Proc. Amer. Math. Soc. 130 (2002), 9-14. MR 1855613 (2002g:11186)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|