Tenth order mock theta functions in Ramanujan's lost notebook (IV)
Author:
YounSeo Choi
Journal:
Trans. Amer. Math. Soc. 354 (2002), 705733
MSC (2000):
Primary 11B65; Secondary 11F20, 33E05
Published electronically:
September 21, 2001
MathSciNet review:
1862564
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Abstract: Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.
 [AH]
George
E. Andrews and Dean
Hickerson, Ramanujan’s “lost” notebook. VII. The
sixth order mock theta functions, Adv. Math. 89
(1991), no. 1, 60–105. MR 1123099
(92i:11027), http://dx.doi.org/10.1016/00018708(91)90083J
 [B1]
Bruce
C. Berndt, Ramanujan’s notebooks. Part III,
SpringerVerlag, New York, 1991. MR 1117903
(92j:01069)
 [B2]
Bruce
C. Berndt, Ramanujan’s notebooks. Part IV,
SpringerVerlag, New York, 1994. MR 1261634
(95e:11028)
 [BR]
Bruce
C. Berndt and Robert
A. Rankin, Ramanujan, History of Mathematics, vol. 9,
American Mathematical Society, Providence, RI; London Mathematical Society,
London, 1995. Letters and commentary. MR 1353909
(97c:01034)
 [C1]
YounSeo
Choi, Tenth order mock theta functions in Ramanujan’s lost
notebook, Invent. Math. 136 (1999), no. 3,
497–569. MR 1695205
(2000f:11016), http://dx.doi.org/10.1007/s002220050318
 [C2]
Y.S. Choi, Tenth order mock theta functions in Ramanujan's Lost Notebook (II), Adv. Math. 156 (2000), 180285. CMP 2001:07
 [C3]
Y.S. Choi, Two identities for tenth order mock theta functions in Ramanujan's Lost Notebook, submitted for publication.
 [GR]
George
Gasper and Mizan
Rahman, Basic hypergeometric series, Encyclopedia of
Mathematics and its Applications, vol. 35, Cambridge University Press,
Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
(91d:33034)
 [H1]
Dean
Hickerson, A proof of the mock theta conjectures, Invent.
Math. 94 (1988), no. 3, 639–660. MR 969247
(90f:11028a), http://dx.doi.org/10.1007/BF01394279
 [H2]
Dean
Hickerson, On the seventh order mock theta functions, Invent.
Math. 94 (1988), no. 3, 661–677. MR 969248
(90f:11028b), http://dx.doi.org/10.1007/BF01394280
 [KM]
Marvin
I. Knopp, Modular functions in analytic number theory, Markham
Publishing Co., Chicago, Ill., 1970. MR 0265287
(42 #198)
 [ML]
L. J. Mordell, The value of the definite integral, Quarterly Journal of Mathematics 48 (1920), 329342.
 [RA]
Srinivasa
Ramanujan, The lost notebook and other unpublished papers,
SpringerVerlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an
introduction by George E. Andrews. MR 947735
(89j:01078)
 [RR]
Robert
A. Rankin, Modular forms and functions, Cambridge University
Press, CambridgeNew YorkMelbourne, 1977. MR 0498390
(58 #16518)
 [RS]
Sinai
Robins, Generalized Dedekind 𝜂products, The
Rademacher legacy to mathematics (University Park, PA, 1992) Contemp.
Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994,
pp. 119–128. MR 1284055
(95k:11061), http://dx.doi.org/10.1090/conm/166/01645
 [WG]
G. N. Watson, The final problem:An account of the mock theta functions, J. London Math. Soc. 11 (1936), 5580.
 [AH]
 G. E. Andrews and D. Hickerson, Ramanujan's "Lost" Notebook VII:The sixth order mock theta functions, Adv. Math. 89 (1991), 60105. MR 92i:11027
 [B1]
 B. C. Berndt, Ramanujan's Notebooks Part III, SpringerVerlag, New York, 1991. MR 92j:01069
 [B2]
 B. C. Berndt, Ramanujan's Notebooks Part IV, SpringerVerlag, New York, 1994. MR 95e:11028
 [BR]
 B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., Providence, 1995; London Math. Soc., London, 1995. MR 97c:01034
 [C1]
 Y.S. Choi, Tenth order mock theta functions in Ramanujan's Lost Notebook, Invent. Math. 136 (1999), 497569. MR 2000f:11016
 [C2]
 Y.S. Choi, Tenth order mock theta functions in Ramanujan's Lost Notebook (II), Adv. Math. 156 (2000), 180285. CMP 2001:07
 [C3]
 Y.S. Choi, Two identities for tenth order mock theta functions in Ramanujan's Lost Notebook, submitted for publication.
 [GR]
 G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. MR 91d:33034
 [H1]
 D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639660. MR 90f:11028a
 [H2]
 D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), 661677. MR 90f:11028b
 [KM]
 M. I. Knopp, Modular Functions in Analytic Number Theory, reprint, Chelsea Publishing Co., New York, 1993. MR 42:198 (original ed.)
 [ML]
 L. J. Mordell, The value of the definite integral, Quarterly Journal of Mathematics 48 (1920), 329342.
 [RA]
 S. Ramanujan, The Lost Notebook and Other Unpublished papers, Narosa Publishing House, New Delhi, 1988. MR 89j:01078
 [RR]
 R. A. Rankin, Modular Forms and Functions, Cambridge University Press, Cambridge, 1977. MR 58:16518
 [RS]
 S. Robins, Generalized Dedekind products, Contemp. Math. 166 (1994), 119128. MR 95k:11061
 [WG]
 G. N. Watson, The final problem:An account of the mock theta functions, J. London Math. Soc. 11 (1936), 5580.
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Additional Information
YounSeo Choi
Affiliation:
Department of Mathematics, Korea University, 51, Anamdong, Sungbukku, Seoul, 136701, Korea
Email:
ychoi2@mail.korea.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002994701028616
PII:
S 00029947(01)028616
Keywords:
Ramanujan,
definite integral,
theta function,
mock theta function
Received by editor(s):
August 11, 2000
Published electronically:
September 21, 2001
Article copyright:
© Copyright 2001
American Mathematical Society
