I’m interested in certain generating functions of Beatty sequences like the following: $$f_\alpha (x)=\sum_{n\ge 1} x^n \lfloor n\alpha\rfloor$$ $$g_\alpha (x,y)=\sum_{n\ge 1} x^n y^{\lfloor n\alpha\rfloor}$$ where $\alpha$ is an irrational number. I’d like to know if there are any nontrivial special values or functional equations known for these functions (in particular, I’m looking for results that aren’t direct translations of theorems from number theory into GFs). Can anyone suggest any papers/online resources with actual calculated special values for these functions or nontrivial functional equations?
I already know the following functional equations: $$f_\alpha(x)=g_{1/\alpha}(1,x)$$ $$\frac{1-x}{x}g_\alpha(x,y)+\frac{1-y}{y}g_{1/\alpha}(y,x)=1$$ Despite these nice and pleasingly symmetrical functional equations, I haven’t been able to work out any nontrivial special values of these functions.
I already know about this paper connecting Beatty sequences to generating functions. However, it does not suit my purposes because it uses GFs as a technique to prove results about Beatty sequences, whereas I’m interested in explicitly calculating special values of the generating functions.