On the average sum of the $k$-th divisor function over values of quadratic polynomials

Let $F(\mathbf{x})\in \mathbb{Z}[x_1,x_2,\dots ,x_n]$ be a quadratic polynomial in $n\ge 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $${\mathrm{\Sigma }}_{k,F}(X;\mathcal{B})=\sum _{\mathbf{x}\in X\mathcal{B}\cap {\mathbb{Z}}^n}{\tau }_k(F(\mathbf{x})),$$ for $X$ tending to infinity, where $\mathcal{B}\subset {\mathbb{R}}^n$ is an $n$-dimensional box such that $\min\limits_{{\bf x}\in X$\mathcal{B}$}F({\bf x})\ge 0$forallsufficientlylarge$X$,and$\tau_{k}(\cdot)$isthe$k$-th divisor function for any integer $k\ge 2$.