Richard McIntosh

Affiliation: University of Regina

Email: mcintosh@math.uregina.ca

Title Of Talk: A relation between the universal mock theta function $g_2$ and Zwegers' mu-function

Abstract: In this talk I will use the modern notation $a=e^{2\pi iu}$, $b=e^{2\pi iv}$ and $q=e^{2\pi i\tau}$, where $u$ and $v$ are called elliptic variables and $\tau$ is called the modular variable. S.-Y. Kang proved that $$ iag_{2}(a,q)={\eta^4(2\tau)\over\eta^2(\tau)\vartheta(2u\,;2\tau)} +aq^{-1/4}\mu(2u,\tau\,;2\tau)\,, $$ where the Gordon-McIntosh universal mock theta function $g_{2}$ is given by $$ g_{2}(x,q)={1\over j(q,q^2)}\sum_{n=-\infty}^\infty{(-1)^nq^{n(n+1)} \over 1-xq^n} $$ and Zwegers' $\mu$-function is defined by $$ \mu(u,v\,;\tau)=\mu(a,b,q)={A_{l}(a,b,q)\over\vartheta(b,q)}\,. $$ The level $k$ Appell-Lerch function is defined by $$ A_{k}(u,v\,;\tau)=A_{k}(a,b,q)=a^{k/2}\!\!\sum_{n=-\infty}^\infty\!\! {(-1)^{kn} q^{kn(n+1)/2}b^n\over1-aq^n}\,. $$ I will prove that the $\vartheta$-quotient can be removed from Kang's identity to obtain $$ g_{2}(a,q)=-iq^{-1/4}\mu(u,\tau-u\,;2\tau)\,. $$ Generalizations to higher level Appell-Lerch functions will be discussed.

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