Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good name (e.g. generalised polynomial or near-polynomial don't sound so good), and no one I've talked to knows any standard terminology, so I thought I'd ask here.

Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good name (e.g. generalised polynomial or near-polynomial don't sound so good), and no one I've talked to knows any standard terminology, so I thought I'd ask here.

Edit: the motivation for this is the following: if you have a polynomial solution $f(z)$ to a hypergeometric differential equation (i.e. some $_pF_q$ that is a polynomial), then $x^\alpha f(x)$ (for a certain given multiindex $\alpha$) is a solution to the associated GKZ $A$-hypergeometric system in $p+q$ variables. I'm currently working on generalising some results on polynomial solutions to the hypergeometric equation, and I'd like a good name for this kind of power-times-polynomial thing.