Higher order divisor functions over values of mixed powers

Let ${\tau }_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$\sum _{\genfrac{}{}{0px}{}{1\le n_1,n_2,\dots ,n_{\mathrm{\ell }}\le X^{\frac{1}{r}}}{1\le n_{\mathrm{\ell }+1}\le X^{\frac{1}{s}}}}{\tau }_k(n_1^r+n_2^r+\cdots +n_{\mathrm{\ell }}^r+n_{\mathrm{\ell }+1}^s),$$ where $k\ge 4$, $r\ge 2$, $s\ge 2$ and $\mathrm{\ell }\ge 2$ are integers. Previously only special cases are studied.