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A proof of the mock theta conjectures. (English) Zbl 0661.10059

In his “lost” notebook, Ramanujan stated identities involving each of the ten fifth order mock theta functions, which he divided into two groups. G. E. Andrews and F. G. Garvan [Ramanujan’s “Lost” Notebook VI: The mock theta conjectures; Adv. Math. 73, No.2, 242-255 (1989)] showed that these identities are all equivalent to two identities, which they call the first and second mock theta conjectures (one conjecture for each group of mock theta functions). The author proves both of these conjectures. The proof relies on a pair of Hecke type identities discovered by G. E. Andrews [Trans. Am. Math. Soc. 293, 113-134 (1986; Zbl 0593.10018)].
Reviewer: T.M.Apostol

MSC:

11P81 Elementary theory of partitions

Citations:

Zbl 0593.10018
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References:

[1] [A1] Andrews, G.E.: Applications of basic hypergeometric functions. SIAM Rev.16, 441-484 (1974) · Zbl 0299.33004
[2] [A2] Andrews, G.E.: An introduction to Ramanujan’s ?Lost? Notebook. Am. Math. Mon.86, 89-108 (1979) · Zbl 0401.01003
[3] [A3] Andrews, G.E.: Hecke modular forms and the Kac-Peterson identities. Trans. Am. Math. Soc.283, 451-458 (1984) · Zbl 0545.10016
[4] [A4] Andrews, G.E.: The fifth and seventh order mock theta functions. Trans. Am. Math. Soc.293, 113-134 (1986) · Zbl 0593.10018
[5] [A5] Andrews, G.E.: Ramanujan’s fifth order mock theta functions as constant terms. In: ?Ramanujan Revisited: Proc. of the Centenary Conference, Univ. of Illinois at Urbana-Champaign, June 1-5, 1987?. San Diego: Acad. Press 1988 · Zbl 0646.10018
[6] [A-A] Andrews, G.E., Askey, R.: A simple proof of Ramanujan’s summation of the1-1. Aequationes Math.18, 333-337 (1978) · Zbl 0401.33002
[7] [A-G] Andrews, G.E., Garvan, F.G.: Ramanujan’s ?Lost? Notebook VI: The mock theta conjectures, Dept. of Math. Research Report no. 87007, The Pennsylvania State Univ., University Park, PA 16802; to appear in Adv. Math. · Zbl 0677.10010
[8] [A-S] Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. London Math. Soc. (3)4, 84-106 (1954) · Zbl 0055.03805
[9] [G] Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7, and 11. Trans. Am. Math. Soc.305, 47-77 (1988) · Zbl 0641.10009
[10] [H-W] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 4th edition. London: Oxford Univ. Press 1968
[11] [R1] Ramanujan, S.: Collected Papers. Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M. (eds.): London: Cambridge Univ. Press 1927; repritned New York: Chelsea Publ. Cy., 1962
[12] [R2] Ramanujan, S.: The lost notebook and other unpublished papers. New Delhi: Narosa Publishing House 1988 · Zbl 0639.01023
[13] [S] Selberg, A.: Über die Mock-Thetafunktionen siebenter Ordnung. Arch. Math. Naturvidenskab41, 3-15 (1938) · JFM 64.1091.03
[14] [W1] Watson, G.N.: The final problem: an account of the mock theta functions. J. London Math. Soc.11, 55-80 (1936) · Zbl 0013.11502
[15] [W2] Watson, G.N.: The mock theta functions (2). Proc. London Math. Soc. (2)42, 274-304 (1937) · Zbl 0015.30402
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