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?



Relation between Legendre and Chebyshev polynomials
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].


