On the rank of Hankel matrices over finite fields

Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ over $F$ have rank $\leq r$ ? This question is classical, the answer ($q^{2r}$ when $r\leq\min\left\{ p,q\right\} $) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade Ram). In this note, we study refinement of result: We show that if fix first $k$ entries $x_{0},x_{1},\ldots,x_{k-1}$ for some $k\leq r\leq\min\left\{ $, then number ways to choose remaining $p+q-k+1$ $x_{k},x_{k+1},\ldots,x_{p+q}$ such resulting matrix $q^{2r-k}$. exactly one would expect had no effect on rank, but course situation not simple. The refined result generalizes (and provides an alternative proof of) Anzis, Chen, Gao, Kim, Li Patrias evaluations Jacobi-Trudi determinants fields.