Let $p(z):= a_0 + a_1z + \cdots + a_nz^n$ be a degree $n$ polynomial, let $m$ be a large integer (which we may assume much larger than $n$), and let $k$ be some integer in the range $m+1, \cdots, m+n$. I am trying (for a paper I am writing) to find an upper bound for the value
$$c_{m,k}:= \sum_{j=\max\{0, k-n\}}^m a_jb_{k-j}$$
that results from taking coefficient $[z^k]$ of the product $p\cdot \left(\frac{1}{p} * \frac{1-z^{k+1}}{1-z}\right)$ (the symbol $*$ meaning the coefficient-wise Hadamard product). It is clear that this sum is asymptotically $\rho^{-k}\theta(k)$, where $\rho:=\min_{p(r) = 0} \{\left|r\right|\}$ and $\theta(k)$ is subexponential. However, I am trying to find the subexponential term $\theta(k)$ and, having tried nearly everything, am badly failing. In particular,
- it's not at all clear from most texts on recurrence relations how the term $b_j$ should behave apart from being asymptotically $\rho^{-j}$;
- the terms $c_{m,k}$ are the coefficients of a polynomial sequence formed by recurrence relations, though not one that seems to be orthogonal or fit into any well-studied category of such sequences;
- there exist a few integral expansions for Hadamard products, none of which seems to reduce the problem to an easier one;
- this quantity is equal to the integral $$\frac{1}{2\pi i} \int_{\gamma} \frac{p_{k-m-1}(z)}{z^k\cdot p(z)}\;dz,$$ where $p_j(z):=$ the $j^{th}$ degree Taylor approximation of $p(z)$ and $\gamma$ is the doughnut formed by the unit circle taken anticounterclockwise and a circle of radius smaller than $\rho$ taken clockwise, BUT only the Residue Theorem seems useful to calculate this integral, essentially putting me back at square one;
- I have a very weak upper bound on $b_j$ that (1) would force me to use the Triangle Inequality and (2) depends on the order of the zeros and moduli of residues anyway.
Please help! I've spent way too long on this silly thing and my adviser is understandably on my butt about it.