The sum of a hybrid arithmetic function over a sparse sequence

Let ${\lambda }_f(n)$ be the $n$-th normalized Fourier coefficient of $f$, which is a primitive holomorphic cusp form of even integral weight $k\ge 2$ for the full modular group $S{L}_2(\mathbb{Z})$. Let also $\sigma (n)$ and $\varphi (n)$ be the sum-of-divisors function and the Euler totient function, respectively. In this paper, we are able to establish the asymptotic formula of the sum of the hybrid arithmetic function ${\lambda }_f^l(n){\sigma }^c(n){\varphi }^d(n)$ over the sparse sequence $\{n:n=a^2+b^2\}$, namely, $\sum _{n\le x}{\lambda }_f^l(n){\sigma }^c(n){\varphi }^d(n)r_2(n)$ for $1\le l\le 8$, where $x$ is a sufficiently large real number, the function $r_2(n)$ denotes the number of representations of $n$ as $n=a^2+b^2$, $a,b,l\in \mathbb{Z}$ and $c,d\in \mathbb{R}$.