ASYMPTOTIC EXPANSIONS OF ZEROS OF A PARTIAL THETA FUNCTION
Résumé
The bivariate series θ(q, x) := ∞ j=0 q j(j+1)/2 x j defines a partial theta function. For fixed q (|q| < 1), θ(q, .) is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in q of the zeros of θ. The coefficients r k of the stabilized series are positive integers. They are the elements of a known increasing sequence satisfying the recurrence relation r k = ∞ ν=1 (−1) ν−1 (2ν + 1)r k−ν(ν+1)/2 .