In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta functions, because as $q$ radially approaches any point $e^{2\pi ir}$ ($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$. In this paper we establish the relationship between two families of mock theta functions.

Keywords:

$q$-series, mock theta function, Mordell integral $q$-series, mock theta function, Mordell integral

MSC Classifications:

11B65, 33D15 show english descriptions Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
Basic hypergeometric functions in one variable, ${}_r\phi_s$ 11B65 - Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
33D15 - Basic hypergeometric functions in one variable, ${}_r\phi_s$

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