On the double zeros of a partial theta function

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1)q∈[0,1), x∈Rx∈R, and defines a partial theta function  . For any fixed q∈(0,1)q∈(0,1) it has infinitely many negative zeros. For q taking one of the spectral   values q˜1, q˜2, … (where 0.3092493386…=q˜1