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Working with the definition of Hermite polynomials

$x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then

$$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} [f'(x_j)\hat{H}_{n,j}(x)],$$


$H_{n,j}(x) = [1-2(x-x_{j})L'_{n,j}(x)]L_{n,j}^2(x)$,


and $L_{n,j}(x)$ is the $j$th Lagrange coefficient polynomial of degree n.

Is it possible to extend this definition to $H_{3n+2}$?

If so, what is the relationship between $H_{n,j}$ and $\hat{H}_{n,j}$ and how can we find $\hat{\hat{H}}_{n,j}$ such that

$$H_{3n+2}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} [f'(x_j)\hat{H}_{n,j}(x)] +\sum_{j=0}^{n} [f''(x_j)\hat{\hat{H}}_{n,j}(x)]?$$

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