I am interested in the following integral related to the Chebyshev polynomials


$$I\_{n,m}:= \int\_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$


where $n,m\in \mathbb{Z}^+.$


It is easy to see the following result.


For an even number $n \in \mathbb{Z}^+$ and an odd number $m \in \mathbb{N}$, we have


$$I\_{n,m}=0.$$


The nonzero $I\_{n,m}$ are the so called central multinomial coefficients , the largest coefficient of $(1+x+x^2\cdots +x^{n-1})^m$.


I conjecture that if $I\_{n,m}\neq 0$, then the $I\_{n,m}$ is a polynomial $P(n)$ of degree $m-1$. 


For example, the following results have been proved :


$\displaystyle I\_{n,1} = \pi\enspace$ for odd $\,n\,$, otherwise $\,0\,$;


$\displaystyle I\_{n,2} = \pi n$;


$\displaystyle I\_{n,3} =\frac{\pi}{4}(1+3n^2)\enspace$ for odd $\,n\,$, otherwise $\,0\,$;


$\displaystyle I\_{n,4}= \frac{\pi n}{3}(1+2n^2)$.


But in the general case, I have no idea about the proof of the conjecture. If this conjecture is true, then for $\forall n,m\in \mathbb{Z}^+ $, we can determine $I\_{n,m}$ by the method of interpolation.


If someone can give some suggestion or opinion on the proof of the conjecture, I will appreciate it.



A Conjecture about the integral related to Chebyshev polynomial
This is proven in [arXiv:1002.3844](https://arxiv.org/abs/1002.3844), see top of page 12. The quantity $A\_k^b(n)$ in that paper is a polynomial in $n$ of degree $\leq k$ and it is related to the integral $I\_{nm}=\pi P\_m(n)$ in the OP by $A\_{k}^b(n)=P\_{k+1}(2n+1)$.


The polynomial can be expressed as a terminating hypergeometric series, see equation 83. I also note that, according to remark 3 on page 13, the polynomial $P\_m(n)$ is even for $m$ odd and odd for $m$ even.


