Chebyshev polynomials of the second kind $V\_n(x)$ can be defined as
$$V\_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$
or through the recurrence relation
$$V\_{n+1}=xV\_n-V\_{n-1}, V\_0=1, V\_1=x.$$
First few low-order Chebyshev polynomials of the second
kind are as follows:
$$V\_0=1, V\_1=x,V\_2=x^2-1,V\_3=x^3-2x,$$
$$V\_4=x^4-3x^2+1,V\_5=x^5-4x^3+3x.$$


I want to know how to calculate the following integral relate to Chebyshev polynomials:
$$\int\_0^\pi (\frac{\sin nx}{\sin x})^m dx$$
where $n,m\in \mathbb{Z}^+$.


It is easy to see the following result:


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


$$ \int\_0^\pi (\frac {\sin nx}{\sin x})^{m} dx=0.$$


I conjectured that the result is a polynomial $P(n)$ of order $m−1$. But I have no idea about the proof. 


I prefer to know the other two cases beside the above special case. Thank you.



how to calculate the following integral related to Chebyshev polynomials
$$I\_{nm}=\frac{1}{\pi}\int\_0^\pi \left(\frac{\sin nx}{\sin x}\right)^m dx$$


As stated in the OP, $I\_{nm}=0$ for $n$ even and $m$ odd. 
Otherwise, the nonzero $I\_{nm}$ is the *central multinomial coefficient* $c\_{nm}$, the largest coefficient of $(1+x+x^2\cdots x^{n-1})^m$.


A few examples:
$$I\_{44}=44\;\; \text{and}\;\; (1 + x + x^2 + x^3)^4=1 + \cdots + 40 x^5 + \mathbf{44}\, x^6 + 40 x^7 + 
 \cdots + x^{12}$$
$$I\_{34}=19\;\; \text{and}\;\; (1 + x + x^2 )^4=1 + \cdots + 16 x^3 + \mathbf{19}\, x^4 + 16 x^5 + 
 \cdots + x^{8}$$
$$I\_{33}=7\;\; \text{and}\;\; (1 + x + x^2 )^3=1 + 3 x + 6 x^2 + \mathbf{7}\, x^3 + 6 x^4 + 3 x^5 + x^6$$


For small $n$ the central multinomial coefficients are listed on [OEIS](https://oeis.org), for example, $I\_{6m}$ is given for $m=2,4,6,8,\ldots$ by A063419:


$$ 6, 146, 4332, 135954, 4395456, 144840476, 4836766584, 163112472594, 5542414273884, 189456975899496, 6507792553644256, \ldots$$


An efficient method to calculate these coefficients is described in [On a link between Dirichlet kernels and central multinomial coefficients.](https://www.sciencedirect.com/science/article/pii/S0012365X15001260)




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**Update 1**, following the conjecture added in a comment, that $I\_{nm}=P\_m(n)$ is a polynomial in $n$ of degree $m-1$; I have no proof, but for the record, here is the evidence for small $m$:


$$P\_1(n)=1$$
$$P\_2(n)=n$$
$$P\_{3}(n)=\frac{1}{4} \left(3 n^2+1\right)$$
$$P\_4(n)=\frac{1}{3} n\left(2 n^2+1\right)$$
$$P\_{5}(n)=\frac{1}{192} \left(115 n^4+50 n^2+27\right)$$
$$P\_6(n)=\frac{1}{20} n \left(11 n^4+5 n^2+4\right)$$
$$P\_7(n)=\frac{7 \left(841 n^6+385 n^4+259n^2\right)+1125}{11520}$$
$$P\_8(n)=\frac{1}{315} n \left(151 n^6+70 n^4+49 n^2+45\right)$$
$$P\_9(n)=\frac{259723 n^8+121380 n^6+84882 n^4+64580 n^2+42875}{573440}$$
$$P\_{10}(n)=\frac{1}{36288}n \left(15619 n^8+7350 n^6+5187 n^4+4100 n^2+4032\right)$$


For fixed *even* $m$, these formulas represent closed form expressions for the central multinomial coefficients $c\_{nm}$. On OEIS I find that such polynomial expressions are noted by R.H. Hardin (see  [A071816](https://oeis.org/A071816) and [A005900](https://oeis.org/A005900)), but without a reference to a general proof.


For fixed *odd* $m$, only the $c\_{nm}$'s with odd $n$ follow. I have checked that there is no degree $m-1$ polynomial expressions for the full set of central multinomial coefficients with odd $m$.




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**Update 2**
I have now located a proof in the literature, see <https://mathoverflow.net/a/313854/11260>


