Consider a random scalar variable $X$ with arbitrary measure.
I'm after a basis of polynomial functions $\{p\_k\}\_{k=0}^\infty$ which are orthonormal with respect to $X$ in the sense that


\begin{equation}
\mathbb{E}\_X [p\_k(X)p\_{k'}(X)] = \delta\_{kk'}.
\end{equation}


When discussing orthogonal polynomial bases, the measure of integration is usually assumed.
For example, if $X \sim \mathcal{N}(0,1)$, then $\{p\_k\}\_{k=0}^\infty$ are the Hermite polynomials.
However, it seems there ought to exist generic expressions for such orthogonal polynomials, with the coefficients given in terms of moments of $X$. For example, applying the typical Gram-Schmidt procedure, one can quickly find that


\begin{align}
p\_0(X) &= 1 \\
p\_1(X) &= \frac{X - \mathbb{E}[X]}{\sqrt{\text{Var}[x]}} \\
p\_2(X) &= \ \ \ ...
\end{align}


Are there known expressions for the rest of this polynomial basis (or even just the next few elements)?
In light of the expression for $p\_1$, perhaps centered moments or cumulants are involved.



Orthogonal polynomials w.r.t. an arbitrary measure
For example, you can write orthogonal polynomials as determinants


$p\_n(x) = c\_n \, \det \begin{bmatrix}
m\_0 & m\_1 & m\_2 &\cdots & m\_n \\
m\_1 & m\_2 & m\_3 &\cdots & m\_{n+1} \\
\vdots&\vdots&\vdots&\ddots& \vdots \\
m\_{n-1} &m\_n& m\_{n+1} &\cdots &m\_{2n-1}\\
1 & x & x^2 & \cdots & x^n
\end{bmatrix}$,


where $c\_n$ is some constant for normalization and $m\_k$ is the k-th moment.


A good book concerning orthogonal polynomials is Akhiezer, The Classical Moment problem


