So far, I have only seen examples like the following:
$$ \langle p, q \rangle = \int_0^1 p(x)q(x)\ dx, $$
where $p$ and $q$ are elements (polynomials) of a finite-dimensional polynomial vector space.
I'm wondering if there other kinds of inner products involving polynomials, without involving integrals or more interesting ones of the same kind.
On
The product $\langle p|q\rangle=\sum_n\alpha_n\beta_n$ can be viewed in a fancy way setting the map
$|p\rangle=\sum_n\alpha_nx^n\rightarrow\langle p|=\sum_n\frac{\alpha_n}{n!}\left.\frac{d^n}{dx^n}\right|_{x=0}$
Then,
$\langle p|q\rangle=\sum_{n,m}\frac{\alpha_n\beta_m}{n!}\underbrace{\left.\frac{d^nx^m}{dx^n}\right|_{x=0}}_{n!\delta_{n,m}}=\sum_n \alpha_n\beta_n$.
This is clearly a map between the vector space and its dual,
$x^n\rightarrow \frac{1}{n!}\left.\frac{d^n}{dx^n}\right|_{x=0}$ and thus the basis $\left\lbrace{1,x,x^2,...}\right\rbrace$ is orthonormal.
Every finite-dimensional real vector space can be given an inner product by identifying the space with $\mathbb R^n$ by choosing a basis and transporting the canonical inner product.
For a finite-dimensional polynomial vector space, this gives $$ \langle p, q \rangle = p_0 q_0 + p_1 q_1 + \cdots + p_n q_n $$ when you use the monomial basis $1, x, x^2, \dots, x^n$. Here $p_i$ and $q_i$ are the coefficients of $p$ and $q$. (They are the coordinates with respect to the monomial basis.)