Given a unitary matrix $U$ such that $U^{\dagger} U = UU^{\dagger} = I$, I would like to examine a finite power series $V = \sum\limits_{i=0}^n\alpha_i U^{i}$ where $\alpha_i \neq 0$. Can this operator $V$ be unitary for appropriate choices of $\alpha_i$?
For Pauli matrices, this seems to be possible. For instance if $U = X =\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$, then we know that odd powers of $X$ equal $X$ and even powers equal the identity matrix. One can then rewrite the power series to be $V = \frac{1}{\sqrt{2}}(I + iX)$ while still ensuring all $\alpha_i\neq 0$ and satisfy the requirement that $V$ is unitary.
Is there an extension of such an idea to an arbitrary unitary or at least higher dimensional Pauli matrices?