On average behavior of a hybrid arithmetic function associated to Hecke eigenvalues of cusp forms over the values of a multivariate polynomial

In this paper, we investigate the average behavior of a hybrid arithmetic function, i.e. [Formula: see text] over a certain sequence of positive integers, where [Formula: see text] is the [Formula: see text] normalized Fourier coefficients of the [Formula: see text] ([Formula: see text] is any fixed integer) symmetric power [Formula: see text]-function (i.e. [Formula: see text]), [Formula: see text] and [Formula: see text] are the sum of divisors function and the Euler totient function, respectively. Precisely, we prove an asymptotic formula with an error term for the sum [Formula: see text] where [Formula: see text] is sufficiently large, and [Formula: see text]