I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as this might be important for my problem.
They appeared when calculating the Taylor series of the implicit function $s(x) = 1 - \mu\beta x s(x)^{\beta}$ at $x = 0$ for some parameters $0 < \mu$, $0 <\beta < 1$. I wasn't able to find this explicit form by direct calculation, but Maple gave a very clean expression: $$s(x) = 1 - \mu\beta x + \sum_{k=2}^{n} \frac{1}{k!}\left[\prod_{j=0}^{k-2}(k\beta - j)\right] (-\mu\beta x)^{k}.$$ Any help would be very much appreciated.