For any given integers $m,n\geq1$, let $\mathbb{R}[x\_1,\cdots,x\_n]\_m$ be the vector space of homogeneous polynomials of degree $m$ in $x\_1,\cdots,x\_n$ over the field of real numbers $\mathbb{R}$. Denote by $\mathbb{N}^n\_m$ the following set
$$\mathbb{N}^n\_m=\{(a\_1,\cdots,a\_n)\in \mathbb{Z}^n:a\_1,\cdots,a\_n\geq0,\ a\_1+\cdots+a\_n=m\}.$$
For any $a=(a\_1,\cdots,a\_n)\in \mathbb{N}^n\_m$, denote by $x^a$ the monomial
$$x^a=x\_1^{a\_1}\cdots x\_n^{a\_n}.$$
It is clear that $x^a$ with $a\in \mathbb{N}^n\_m$ form a basis of the vector space $\mathbb{R}[x\_1,\cdots,x\_n]\_m$.


Define an inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}[x\_1,\cdots,x\_n]\_m$ such that for any $a=(a\_1,\cdots,a\_n),b=(b\_1,\cdots,b\_n)\in \mathbb{N}^n\_m$,
$$\langle x^a,x^b\rangle=\left\{
\begin{aligned}
&0& a\neq b\\
&a\_1!\cdots a\_n!\ \ & a=b
\end{aligned}
\right.$$
For any $f(x\_1,\cdots,x\_n),g(x\_1,\cdots,x\_n)\in \mathbb{R}[x\_1,\cdots,x\_n]\_m$, a general expression of $\langle f,g\rangle$ I can think out is as follow:
$$\langle f,g\rangle=\int\_0^{\infty}\cdots\int\_0^{\infty}\int\_0^{2\pi}\cdots\int\_0^{2\pi}e^{-(x\_1+\cdots+x\_n)}
f(\frac{x\_1}{e^{i\theta\_1}},\cdots,\frac{x\_n}{e^{i\theta\_n}})g(e^{i\theta\_1},\cdots,e^{i\theta\_n})dx\_1\cdots dx\_nd\theta\_1\cdots d\theta\_n.$$
Is there any better integral expression of $\langle f,g\rangle$ not using complex integration, or is there any general expression of $\langle f,g\rangle$ not only confined to integral expression? Any idea is welcome!



An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
It looks like you have rediscovered the Bargmann-Segal space! Take $n = 1$ for simplicity. Define $BS$ to be the set of analytic functions in $L^2(\mathbb{C},\mu)$ where $\mu$ is $\frac{1}{\pi}e^{-|z|^2}$ times Lebesgue measure. Thus the inner product of $f,g \in BS$ is $$\langle f,g\rangle = \frac{1}{\pi}\int\_{\mathbb{C}} f(z)\overline{g(z)} e^{-|z|^2}\, dz$$ and a short computation shows that the norm of $z^n$ is $\sqrt{n!}$.


So for $n = 1$, regard your polynomials as complex polynomials with real coefficients, and this single formula for inner product works simultaneously for all values of $m$.


The generalization to arbitrary $n$ is straightforward; the formula for inner product doesn't change. See the Wikipedia page on [Bargmann-Segal space](https://en.wikipedia.org/wiki/Segal%E2%80%93Bargmann_space).


