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The fifth and seventh order mock theta functions. (English) Zbl 0593.10018

The third order mock theta functions, introduced and named by S. Ramanujan, are a class of functions similar to but distinct from theta functions and yet whose behaviour under the fundamental transformations of the modular group can be described. The fifth and seventh order mock theta functions, also introduced by Ramanujan, probably have analyzable behaviour under the fundamental transformations. As a first step in this direction, the author expands the known fifth and seventh order functions into Hecke type series. For the fifth order functions, the expansion is achieved very slickly using the technique of Bailey chains [see the author, Pac. J. Math. 114, 267–283 (1984; Zbl 0547.10012)].

MSC:

11F03 Modular and automorphic functions
11F37 Forms of half-integer weight; nonholomorphic modular forms
11P05 Waring’s problem and variants
05A19 Combinatorial identities, bijective combinatorics

Citations:

Zbl 0547.10012
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References:

[1] G. E. Andrews, On basic hypergeometric series, mock theta functions, and partitions. I, Quart. J. Math. Oxford Ser. (2) 17 (1966), 64 – 80. · Zbl 0136.05603
[2] G. E. Andrews, On basic hypergeometric series, mock theta functions, and partitions. II, Quart. J. Math. Oxford Ser. (2) 17 (1966), 132 – 143. · Zbl 0144.25204
[3] George E. Andrews, On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions, Amer. J. Math. 88 (1966), 454 – 490.
[4] George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. · Zbl 0371.10001
[5] George E. Andrews, An introduction to Ramanujan’s ”lost” notebook, Amer. Math. Monthly 86 (1979), no. 2, 89 – 108. · Zbl 0401.01003
[6] George E. Andrews, Connection coefficient problems and partitions, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 1 – 24.
[7] George E. Andrews, Hecke modular forms and the Kac-Peterson identities, Trans. Amer. Math. Soc. 283 (1984), no. 2, 451 – 458. · Zbl 0545.10016
[8] George E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), no. 2, 267 – 283. · Zbl 0547.10012
[9] W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1948), 1 – 10. · Zbl 0031.39203
[10] D. M. Bressoud, Hecke modular forms and \?-Hermite polynomials, Illinois J. Math. 30 (1986), no. 1, 185 – 196. · Zbl 0577.05009
[11] Leila A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. Amer. Math. Soc. 72 (1952), 474 – 500. · Zbl 0047.27902
[12] Erich Hecke, Mathematische Werke, 3rd ed., Vandenhoeck & Ruprecht, Göttingen, 1983 (German). With introductory material by B. Schoeneberg, C. L. Siegel and J. Nielsen. · Zbl 0092.00102
[13] V. G. Kac and D. H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1057 – 1061. · Zbl 0457.17007
[14] S. Ramanujan, Some properties of Bernoulli’s numbers [J. Indian Math. Soc. 3 (1911), 219 – 234], Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 1 – 14. · JFM 42.0460.02
[15] L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
[16] Daniel Shanks, A short proof of an identity of Euler, Proc. Amer. Math. Soc. 2 (1951), 747 – 749. · Zbl 0044.28403
[17] Daniel Shanks, Two theorems of Gauss, Pacific J. Math. 8 (1958), 609 – 612. · Zbl 0084.06003
[18] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.
[19] G. N. Watson, The final problem: an account of the mock theta functions, Ramanujan: essays and surveys, Hist. Math., vol. 22, Amer. Math. Soc., Providence, RI, 2001, pp. 325 – 347.
[20] -, The mock theta functions. II, Proc. London Math. Soc. (2) 42 (1937), 272-304.
[21] -, A note on Spence’s logarithm transcendant, Quart. J. Math. Oxford Ser. 8 (1937), 39-42. · JFM 63.0319.01
[22] D. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2) 53 (1951), 158 – 180. · Zbl 0044.07705
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