Relation between Legendre and Chebyshev polynomials

Question

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?

Answer 1

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].

Answer 2

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.