In this answer a simple ad-hoc inner product for monomials is defined, since they are known to be orthogonal:
if $P = \sum_{n \ge 0} a_n X^n$ and $Q = \sum_{n \ge 0} b_n X^n$, then $$\left\langle P,Q \right\rangle = \sum_{n \ge 0} a_n b_n$$
My question is, is it legitimate to generalize the above to cases where the leading terms in the polynomials are not integer? For example:
if $P = X^\alpha\sum_{n \ge 0} a_n X^n$ and $Q = X^\beta\sum_{n \ge 0} b_n X^n$ with $\alpha,\beta\in\mathbb{C}$ , then $$\left\langle P,Q \right\rangle = \delta(\alpha-\beta)\sum_{n \ge 0} a_n b_n$$ Does this hold true? And if yes, how to prove that it does / where to read up on the proof?