Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions

For an even integer $k\ge 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let ${\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}(n)$ denote the $n^{\text{th}}$ normalized Fourier coefficient of the $j^{\text{th}}$ symmetric power $L$-function $L(s,{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f)$. It has been an interesting problem to study the average behaviour of ${\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*} \sum_{n\leq x} λ_{\rm{sym}^j}^l(n), \end{equation*} for various values of $l$ and $j$. In this paper, we improve and generalize previously known results concerning the sum above for positive integers $l$ and $j$ such that $lj\ge 4$.