  Blaschke factorization allows us to write any holomorphic function $F$ as a
formal series $$ F = a_0 B_0 + a_1 B_0 B_1 + a_2 B_0 B_1 B_2 + \cdots$$ where
$a_i \in \mathbb{C}$ and $B_i$ is a Blaschke product. We introduce a more
general variation of the canonical Blaschke product and study the resulting
formal series. We prove that the series converges exponentially in the
Dirichlet space given a suitable choice of parameters if $F$ is a polynomial
and we provide explicit conditions under which this convergence can occur.
Finally, we derive analogous properties of Blaschke factorization using our new
variable framework.
