[Paper Review] Stabilization of the asymptotic expansions of the zeros of a partial theta function

Vladimir Petrov Kostov|arXiv (Cornell University)|Oct 9, 2015

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper establishes a stabilization property for the asymptotic expansions of zeros of the partial theta function θ(q,x). By introducing a modified series (H_{m,j}) derived from a known generating series r_k, the authors prove that for sufficiently large j and bounded k, the coefficients g_{j,k} in the Laurent expansion of the j-th zero match those of (H_{m,j}), with explicit bounds on j and k where this stabilization holds, using recursive partition counts r_k.

ABSTRACT

The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em 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 $-q^{-j}+(-1)^jq^{j(j-1)/2}(1+\sum _{k=1}^{\infty}g_{j,k}q^k)$. 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 $r_k=\sum _{ν=1}^{\infty}(-1)^{ν-1}(2ν+1)r_{k-ν(ν+1)/2}$. Set $(H_{m,j})~:~(\sum _{k=0}^{\infty}r_kq^k) (1-q^{j+1}+q^{2j+3}-\cdots +(-1)^{m-1}q^{(m-1)j+m(m-1)/2})= \sum _{k=0}^{\infty} ilde{r}_{k;m,j}q^k$. Then for $k\leq (m+2j)(m+1)/2-1-j$ and $j\geq (2m-1+\sqrt{8m^2+1})/2$ one has $g_{j,k}= ilde{r}_{k;m,j}$.

Motivation & Objective

• To refine the known stabilization property of Laurent series coefficients g_{j,k} in the asymptotic expansion of the j-th zero of the partial theta function θ(q,x).
• To establish precise conditions under which the coefficients g_{j,k} match those of a modified generating series (H_{m,j}), ensuring stability across increasing j.
• To characterize the coefficients g_{j,k} using a recurrence based on partition numbers r_k into three distinct kinds of parts.
• To determine the range of k and j where non-linear terms in the expansion do not affect the computation of g_{j,k}, enabling exact matching with (H_{m,j}).

Proposed method

• Define the modified series (H_{m,j}) as the product of the generating series ∑r_k q^k and a finite alternating sum of powers of q, forming ∑r̃_{k;m,j} q^k.
• Use matrix analysis of Laurent expansions of monomials Ψ_ν = q^{ν(ν+1)/2} (-ξ_j)^ν to track coefficient contributions and identify linear systems governing g_{j,k}.
• Establish linear systems (2), (3), and (4) from the matrix structure, where coefficients of g_{j,k} are derived from paired monomials Ψ_ν and Ψ_{2j-1−ν} with weights ν and (2j−1−ν).
• Prove that for k ≤ (m+2j)(m+1)/2 − 1 − j and j ≥ (2m−1 + √(8m²+1))/2, the coefficients g_{j,k} equal r̃_{k;m,j}, ensuring stabilization.
• Use the minimal power of q in non-linear terms, j(j+3)/2, to derive the lower bound on j, ensuring non-linear terms do not interfere with coefficient computation.
• Leverage the known recurrence r_k = ∑_{ν=1}^∞ (−1)^{ν−1}(2ν+1) r_{k−ν(ν+1)/2} with r_0=1 and r_k=0 for k<0, to define the partition-based coefficients r_k.

Experimental results

Research questions

• RQ1 For which values of j and k do the coefficients g_{j,k} in the Laurent expansion of the j-th zero of θ(q,x) stabilize to match those of a modified series (H_{m,j})?
• RQ2 What is the precise range of k for which the coefficients g_{j,k} remain unaffected by non-linear terms in the expansion of θ(q,−ξ_j)?
• RQ3 How does the inclusion of additional terms in the series (H_{m,j}) via successive column additions to a coefficient matrix affect the stability of g_{j,k}?
• RQ4 What role do the partition numbers r_k—counting partitions into three distinct kinds of parts—play in the stabilization of the coefficients g_{j,k}?
• RQ5 Under what conditions on j and m does the absence of higher-order monomials Ψ_ν (ν ≥ m+2j) not affect the computation of g_{j,k}?

Key findings

• The coefficients g_{j,k} in the Laurent expansion of the j-th zero of θ(q,x) match those of the series (H_{m,j}) for all k ≤ (m+2j)(m+1)/2 − 1 − j and j ≥ (2m−1 + √(8m²+1))/2.
• The stabilization condition is valid only when non-linear terms in the expansion of θ(q,−ξ_j) do not contribute to the coefficient of q^k, which occurs when the minimal such power exceeds (m+2j)(m+1)/2.
• The minimal power of q multiplying a non-linear term is j(j+3)/2, and this must be ≥ (m+2j)(m+1)/2 to ensure no interference, leading to the lower bound on j.
• The sequence {r_k} is defined by the recurrence r_k = ∑_{ν=1}^∞ (−1)^{ν−1}(2ν+1) r_{k−ν(ν+1)/2} with r_0=1 and r_k=0 for k<0, and counts partitions into three distinct kinds of parts.
• The first 39 values of r_k are explicitly listed, with r_1=3, r_2=9, r_3=22, r_4=51, r_5=108, r_6=221, r_7=429, r_8=810, r_9=1479, and so on.
• The matrix M_m used in the proof restricts to rows with no non-linear terms, and its column structure ensures that the linear system governing g_{j,k} is equivalent to that of (H_{m,j}) under the stated bounds.