On the zeros of a partial theta function

Abstract

The series θ(q,x):=∑j=0∞qj(j+1)/2xj converges for q∈[0,1), x∈R and defines a partial theta function. For any q∈(0,1) fixed it has infinitely many negative zeros. For countably many values q˜j of q said to form the spectrum of θ (where 0.3092493386…=q˜1<q˜2<⋯<1, limj→∞q˜j=1) the function θ(q,.) has a double zero which is the rightmost of its real zeros (the rest of them being simple). For q≠q˜j it has no multiple real zeros. For q∈(q˜N,q˜N+1) the function θ(q,.) has exactly N complex conjugate pairs of zeros counted with multiplicity (we set q˜0=0). If ξkl denote the zeros of ∂θl/∂xl(q,.) in the order of decreasing, then limk→∞ξklqk=−1 and limk→∞ξk+1l/ξkl=q.