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Tenth order mock theta functions in Ramanujan’s lost notebook. IV. (English) Zbl 1043.33012

Summary: 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.
For Parts I, II see Invent. Math. 136, 497–569 (1999; Zbl 0951.33013), Adv. Math. 156, 180–285 (2000; Zbl 0984.33007).

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
11B65 Binomial coefficients; factorials; \(q\)-identities
11F27 Theta series; Weil representation; theta correspondences
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

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