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I understand that standard Legendre polynomials are normalised such that $$\int_{-1}^1{P_l(x) \cdot P_m(x) ~ dx} = \frac{2}{2l + 1} \delta_{lm}.$$What is the historical reason for this? Does it simplify anything? Who decided on this convention?

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Does it simplify anything? Yes, for example $$ P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}\left(x^{2}-1\right)^{n} $$ and $$ {\frac {1}{\sqrt {1-2xt+t^{2}}}}=\sum _{n=0}^{\infty }P_{n}(x)t^{n} $$ See Wikipedia link.

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Answering your original question with some edit (by myself), this is not a historical issue, it treats only of the usual procedure for normalisation of any set of orthogonal functions. I mean, in mathematics and statistics, given any set of $n$ L.I. functions $f_k(x)$, $k=1,2,\ldots,n$, with which we are intending to form a basis (for the corresponding $n$-dimensional vector-space), the orthonormality condition is just that $\int_a^b{f_m(x) \cdot f_n(x) ~ dx} = \delta_{mn}$.

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  • $\begingroup$ From statistical point of view, this makes sense since a probability should vary from 0 to 1 (i.e., 100 %). $\endgroup$ Sep 15, 2022 at 6:37

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