Let $\omega$ be a positive real number, and define: $$\mathbf{1}_{\omega}\left(n\right)\overset{\textrm{def}}{=}\left(-1\right)^{n}\binom{-\omega}{n}=\binom{\omega-1+n}{n}$$ for all positive integers $n$, and let:$$\zeta_{\omega}\left(s\right)\overset{\textrm{def}}{=}\sum_{n=1}^{\infty}\frac{\mathbf{1}_{\omega}\left(n\right)}{n^{s}}$$ Since $\mathbf{1}_{\omega}\left(1\right)=\omega>0$, $\mathbf{1}_{\omega}$ possesses an inverse with respect to Dirichlet convolution, which I denote by $\mu_{\omega}$, with: $$\sum_{n=1}^{\infty}\frac{\mu_{\omega}\left(n\right)}{n^{s}}=\frac{1}{\zeta_{\omega}\left(s\right)}$$ When $\omega=1$, $\mathbf{1}_{\omega}$ becomes the constant function $1$ and $\mu_{\omega}$ becomes the Möbius function from number theory.
To be clear, this is a reference request: I am looking to see if there has been any work done with these functions. In particular, I am interested in:
• If there is any "official name" for these functions, what is it?
• Closed-form expressions for $\mu_{\omega}\left(n\right)$ (analogous to how the möbius function can be computed in terms of the prime factors of its inputs)
• Asymptotics/formulas for $\mu_{\omega}\left(n\right)$, $\sum_{k=1}^{n}\mu_{\omega}\left(k\right)$ and $\sum_{d\mid n}\mu_{\omega}\left(d\right)$ as $n\rightarrow\infty$.
I'm making tedious progress brute-forcing the formulas by hand, but since this is only tangential to what I'm actually working on, it would be of much help if it turned out that someone had already done that drudgery for me. ;)
