Multidimensional exponential divisor function over Gaussian integers

Let ${\text{taue}}_k\text{\hspace{0.17em}⁣}:\mathbb{Z}\to \mathbb{Z}$ be a multiplicative function such that ${\text{taue}}_k(p^a)={\sum }_{d_1\mathrm{.}\mathrm{.}\mathrm{.}{d}_k=a}1$. In the present paper we introduce generalizations of ${\text{taue}}_k$ over the ring of Gaussian integers $\text{Zi}$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.