STABILIZATION OF THE ASYMPTOTIC EXPANSIONS OF THE ZEROS OF A PARTIAL THETA FUNCTION
Résumé
The bivariate series θ(q, x) := Σ∞ (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. These series are of the form −q −j + (−1) j q j(j−1)/2 (1 + Σ∞(k=1) g j,kq^k). The coefficients of the stabilized series are expressed by the positive integers r k giving the number of partitions into parts of three different kinds. They satisfy the recurrence relation rk = Σ∞ (ν=1) (−1)^(v−1) (2ν + 1)r k−ν(ν+1)/2. Set (H m,j) : Σ∞ (k=0) r k q k (1 − q j+1 + q 2j+3 − · · · + (−1) m−1 q (m−1)j+m(m−1)/2) = Σ∞ (k=0) r k;m,j q k. Then for k ≤ (m + 2j)(m + 1)/2 − 1 − j and j ≥ (2m − 1 + √ (8m^2 + 1))/2 one has g j,k = ˜ r k;m,j .