Hi @Chemicaleng,
Thanks for the post.
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As @nluethen has pointed out in another post, the basis functions you defined here corresponds to a standard random variable. For example, your Legendre polynomials L are mutually orthogonal with respect to the probability measure of the uniform distribution \mathcal{U}(-1,1), whereas the Hermite polynomials you defined form a set of basis with respect to the probability measure of the standard normal distribution \mathcal{N}(0,1). For a simple verification, you calculate some properties of the basis function in your table L^2(X_1) = \frac{1}{2}\left(3\,X_1^2 - 1\right): its expectation should be zero, which is not the case if you use X_1 \sim \mathcal{U}(-1,3).
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Yes. As far as your input variables are independent, you can multiply the associated univariate basis functions to get multivariate bases.
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In general, UQLab does not use the explicit form of the orthogonal polynomials. Instead, they are evaluated efficiently in a recursive way. For arbitrary PCEs, please refer to the PCE manual (Section 1.3.1.2, page 3) and also this book for details.
I hope my answer helps.
Best,
Xujia