The \(H\) and \(K\) families of mock theta functions (Q2903313)

The families of mock theta functions considered in this paper are NEWLINE\[NEWLINE H(x,q) := \sum_{n \geq 0} \frac{q^{n(n+1)/2}(-q;q)_n}{(x;q)_{n+1}(q/x;q)_{n+1}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE K(y,q) := \sum_{n \geq 0} \frac{(-1)^nq^{n^2}(q;q^2)_n}{(yq^2;q^2)_{n}(q^2/y;q^2)_{n}}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE K_1(y,q) := \sum_{n \geq 0} \frac{(-1)^nq^{(n+1)^2}(q;q^2)_n}{(yq;q^2)_{n+1}(q/y;q^2)_{n+1}}, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE K_2(y,q) := \sum_{n \geq 0} \frac{q^{n(n+1)/2}(-1;q)_n}{(yq;q)_{n}(q/y;q)_{n}}, NEWLINE\]NEWLINE where as usual NEWLINE\[NEWLINE (a;q)_n := (1-a)(1-aq)\cdots (1-aq^{n-1}). NEWLINE\]NEWLINE The most important of these is \(H(x,q)\), which is dubbed a ``universal'' mock theta function, since it turns out that all of the classical mock theta functions are (up to the addition of a modular form) specializations of it. The author begins in Section 2 with identities expressing the families of \(q\)-series in terms of Appell-Lerch series. In Section 3 he gives a number of relations among the functions, such as NEWLINE\[NEWLINE \frac{qH(x,q)}{x} + \frac{K(-x^2/q,q^2)}{1+x^2/q} = \frac{(q^2;q^2)_{\infty}^3}{j(x,q)j(-x^2/q,q^4)}. NEWLINE\]NEWLINE Here \(j(x,q) := (x;q)_{\infty}(q/x;q)_{\infty}(q;q)_{\infty}\). In Section 4 he establishes transformation laws for the function \(H(q^r,q)\). In Section 5 he writes classical even-order mock theta functions, up to the addition of a modular form, as specializations of \(H(x,q)\). (He cautions that in the case of two of Ramanujan's tenth-order mock theta functions a proof has yet to be worked out.)