Three-parameter mock theta functions

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Mock theta functions were first introduced by Ramanujan. Historically, mock theta functions can be represented as Eulerian forms, Appell-Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, in view of the q -Zeilberger algorithm and the Watson–Whipple transformation formula, we establish five three-parameter mock theta functions in Eulerian forms, and express them by Appell–Lerch sums. Especially, the main results generalize some two-parameter mock theta functions. For example, setting ( m , q , x ) → ( 1 , q 1 / 2 , x q − 1 / 2 ) in ∑ n = 0 ∞ ( − q 2 ; q 2 ) n q n 2 + ( 2 m − 1 ) n ( x q m , x − 1 q m ; q 2 ) n + 1 , we derive the universal mock theta function g 2 ( x , q ) .