Properties of the recursive divisor function and the number of ordered factorizations

We recently introduced the recursive divisor function ${\kappa }_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is $\zeta (s-x)\mathrm{/}(2-\zeta (s))$. We show that ${\kappa }_x(n)$ is related to the ordinary divisor function by ${\kappa }_x\ast {\sigma }_y={\kappa }_y\ast {\sigma }_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating ${\kappa }_x$ and some standard arithmetic functions. We also clarify the relation between ${\kappa }_0$ and the much-studied number of ordered factorizations $K(n)$, namely, ${\kappa }_0=\mathbf{1}\ast K$.