## Ramanujan’s “Lost” Notebook. VII: The sixth order mock theta functions.(English)Zbl 0739.11042

The authors prove 11 identities from Ramanujan’s lost notebook. These involve seven functions defined by $$q$$-series, which are called mock theta functions. Let \begin{aligned} (x)_ n &=(x;q)_ n =\begin{cases} \prod_{i=0}^{n-1}(1-q^ ix), &n\geq 0,\\ \prod_{i=1}^{-n}(1- q^{-i}x)^{-1}, &n<0,\;x\neq q,q^ 2,\dots\hbox{ or } q^{-n}, \end{cases} \\ (x)_ \infty &=(x;q)_ \infty=\prod_{i\geq 0}(1-q^ i x), \hbox{ where } | q| <1, \\ \phi(q) &=\sum_{n\geq 0} {(- 1)^ n q^{n^ 2}(q;q^ 2)_ n \over (-q)_{2n}}, \\ \psi(q) &=\sum_{n\geq 0} {(-1^ n)q^{(n+1)^ 2}(q;q^ 2)_ n \over (- q)_{2n+1}}, \\ j(x,q)&=\sum_ n(-1)^ nq^{\left({x \atop 2}\right)}x^ n, \quad | q|<1. \\ \end{aligned} Two of the identities are \begin{aligned} \phi(q^ 9)-\psi(q)-q^{-3}\psi(q^ 9) &={{j(- q^ 3,q^{12})(q^ 6;q^ 6)^ 2_ \infty} \over {j(-q,q^ 4)j(- q^ 9,q^{36})}}, \\ {\psi(\omega q)-\psi(\omega^ 2 q)\over (\omega- \omega^ 2)q} &= {{j(-q,q^ 4)j(-q^ 9,q^{36})(q^ 3;q^ 6)_ \infty} \over {j(-q^ 3,q^{12})}} \\ \end{aligned} where $$\omega$$ is a primitive cubic root of 1. Results concerning $$\Theta$$ functions are proved first. Bailey pair method and constant term method are described next. These are used to derive Hecke type identities for the sixth order mock theta functions. The asymptotics of the functions for $$q$$ near a root of unity are also discussed.
Reviewer: M.Cheema (Tucson)

### MSC:

 11P82 Analytic theory of partitions 11F03 Modular and automorphic functions 05A30 $$q$$-calculus and related topics
Full Text:

### References:

 [1] Andrews, G. E., The theory of partitions, (Rota, G.-C, Encyclopedia of Mathematics and Its Applications, Vol. 2 (1976), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0155.09302 [2] Andrews, G. E., Hecke modular forms and the Kac-Peterson identities, Trans. Amer. Math. Soc., 283, 451-458 (1984) · Zbl 0545.10016 [3] Andrews, G. E., Multiple series Rogers-Ramanujan type identities, Pacific J. Math., 114, 267-283 (1984) · Zbl 0547.10012 [4] Andrews, G. E., Combinatorics and Ramanujan’s “lost” notebook, (Anderson, I., Surveys in Combinatorics 1985. Surveys in Combinatorics 1985, London Math. Soc. Lecture Note Series, Vol. 103 (1985), Cambridge Univ. Press: Cambridge Univ. Press London), 1-23 · Zbl 0575.05004 [5] Andrews, G. E., $$q$$-series: Their development and application in analysis, number theory, combinatorics, physics, and computer algebra, (Regional Conference Series in Mathematics, Vol. 66 (1986), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · Zbl 0594.33001 [6] Andrews, G. E., The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293, 113-134 (1986) · Zbl 0593.10018 [7] Andrews, G. E., Ramanujan’s fifth order mock theta functions as constant terms, (Ramanujan Revisited: Proceedings of the Centenary Conference. Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (1988), Academic Press: Academic Press San Diego) · Zbl 0646.10018 [8] Bailey, W. N., Generalized Hypergeometric Series, (Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 32 (1964), Stechert-Hafner Service Agency: Stechert-Hafner Service Agency New York) · Zbl 0011.02303 [9] Bailey, W. N., On the basic bilateral hypergeometric series $$_2ψ_2$$, Quart. J. Math. Oxford Ser., 1, 2, 194-198 (1950) · Zbl 0038.05001 [10] Carlitz, L.; Subbarao, M. V., A simple proof of the quintuple product identity, (Proc. Amer. Math. Soc., 32 (1972)), 42-44 · Zbl 0234.05005 [11] Hickerson, D., A proof of the mock theta conjectures, Invent. Math., 94, 639-660 (1988) · Zbl 0661.10059 [12] Hickerson, D., On the seventh order mock theta functions, Invent. Math., 94, 661-677 (1988) · Zbl 0661.10060 [13] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1968), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0020.29201 [14] Ramanujan, S., (Hardy, G. H.; Aiyar, P. V.Seshu; Wilson, B. M., Collected Papers (1927), Cambridge Univ. Press: Cambridge Univ. Press London) · JFM 53.0030.02 [15] Ramanujan, S., The Lost Notebook and Other Unpublished Papers (1988), Narosa Publishing House: Narosa Publishing House New Delhi · Zbl 0639.01023 [16] Selberg, A., Über. die Mock-Thetafunktionen siebenter Ordnung, Arch. Math. Naturvidenskab, 41, 3-15 (1938) · JFM 64.1091.03 [17] Watson, G. N., The final problem: An account of the mock theta functions, J. London Math. Soc., 11, 55-80 (1936) · Zbl 0013.11502 [18] Watson, G. N., The mock theta functions (2), Proc. London Math. Soc., 42, 2, 274-304 (1937) · Zbl 0015.30402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.