Star product for deformed oscillator algebra $\mathsf{A}\mathsf{q}(2,\nu )$

An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula $\exp[i\epsilon_{\alpha\beta}\partial^\alpha \partial^\beta]$. Using Pochhammer formula, integration over these parameters is carried over a Riemann surface associated with the expression of the type $z^x(1-z{)}^y$ where $x$ and $y$ are arbitrary real numbers.