Sign law for Ramanujan's third order mock theta function $\ddot{I}(q)$

Abstract

We study the coefficients of Ramanujan's third order mock theta function \[ ρ(q)=\sum_{m\geq 0} \frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n. \] Numerical evidence suggests the striking sign pattern \[ r(3n)>0,\qquad r(3n+1)\leq 0,\qquad r(3n+2)\leq 0. \] We prove an asymptotic form of this phenomenon. More precisely, using Watson's relation between $\ddot{I}(q)$ and $\ddot{I}‰(q)$, together with a Rademacher-type expansion for the coefficients of $\ddot{I}‰(q)$ and the corresponding expansion for a theta--eta product, we show that \[ r(n)\sim κ_{n\bmod 3}\, \frac{2π}{(12n+8)^{1/4}} I_{1/2}\!\left(\frac{π\sqrt{12n+8}}{18}\right), \] where \[ κ_0=\frac13\cos\fracπ{18}>0, \qquad κ_1=-\frac13\sin\frac{2π}{9}<0, \qquad κ_2=-\frac13\sin\fracπ{9}<0. \] Consequently, \[ r(3n)>0, \qquad r(3n+1)<0, \qquad r(3n+2)<0 \] for all sufficiently large $n$.