TITLE: Relation between Legendre and Chebyshev polynomials
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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].