  We give a generalization of the notion of finite Blaschke products from the
perspective of generalized inner functions in various reproducing kernel
Hilbert spaces. Further, we study precisely how these functions relate to the
so-called Shapiro--Shields functions and shift-invariant subspaces generated by
polynomials. Applying our results, we show that the only entire inner functions
on weighted Hardy spaces over the unit disk are multiples of monomials,
extending recent work of Cobos and Seco.
