Where I could find relationships between Legendre and Chebyshev polynomials? For example I found with maple $$ P_n(\cos\theta)=\sum_{k=0}^n(-1)^{n+k}\frac{2-\delta_{k0}}{4^n} \binom{n-k}{\frac{n-k}{2}}\binom{n+k}{\frac{n+k}{2}}\cos(k\theta)$$ The sum runs over $n+k$ even, and $\delta_{k0}=1$ if and only if $k=0$. (And $\cos(k\theta)$ are the Chebyshev polynomials)

But would like to know how its proved, and what the inverse relationship is. Are there any papers or books with these types of relationships?


Both the Legendre and Chebyshev polynomials are particular cases of Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$. A general connection formula of the type $$P_n^{(\gamma,\delta)}(x)=\sum_{k=0}^nc_{n,k}^{\gamma,\delta;\alpha,\beta}P_k^{(\alpha,\beta)}(x)$$ can be found on page 256 of the book [Mourad E.H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, Cambridge, 2005].


On pp.13~15 of Fox, L. Parker. Chebyshev polynomials in numerical analysis. No. 519.4 F6. 1968., especially (64)(65), we can see the arguement. As an approach to the minimax solution to the function $\Pi(x)=(x-x_{0})\cdots(x-x_{n})$ with equal weights $w(x)=1$, we can write $$\Pi(x)=\frac{2^{n+1}(n+1)!^{2}}{(2n+2)!}P_{n+1}(x)$$ where $P_{k}(x)$ is a Legendre polynomial of degree $k$.

and by the orthogonal transformation provided there we could also write it in terms of weight $w(x)=\frac{1}{\sqrt{(1-x^{2})}}$ and $$\Pi(x)=2^{-n}T_{n+1}(x)$$ where $T_{k}(x)$ is a Chebyshev polynomial of degree $k$.

And these two equalities gave identity involving both Legendre and Chebyshev polynomials.


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