Would it be possible to define Legendre polynomials in terms of a sinusoidal function for $|x|\leq 1$ in a similar manner to Chebyshev polynomials being defined as $T_n(x) = \cos(n \cos^{-1}(x))$?
Thanks.
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1$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotCommented Aug 27, 2021 at 19:49
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$\begingroup$ Define "similar manner". $\endgroup$– Liviu NicolaescuCommented Aug 27, 2021 at 19:50
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2$\begingroup$ Please don't self-vandalize your post. $\endgroup$– YCorCommented May 23 at 8:36
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1 Answer
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No doubt that $x=\cos\theta$ is a meaningful substitution for the Legendre polynomials. The functions $P_n(\cos\theta)$ were already considered by Legendre in the spherical harmonic expansion of the Newton potential. However, the substitution is not so happy as for the Chebyshev polynomials. Here below is what you get from the generating function of the Legendre's polynomials -not bad after all, but there may be something better. $$P_n(\cos\theta)=4^{-n}\sum_{k=0}^n{2k \choose k}{2n-2k\choose n-k}\cos\big((2n-k)\theta\big).$$
answered Aug 27, 2021 at 22:37
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$\begingroup$ So the above identity can be written as $$P_n = 4^{-n}\sum_{k=0}^n{2k \choose k}{2n-2k\choose n-k}T_{2n-k} $$ which has to be for sure well-known, although I do not find it around right now $\endgroup$ Commented Aug 27, 2021 at 22:45