Simplifying $\sin(nx)$ series notation

Question

I am attempting to simplify the following notation: $$\sin(nx)=\sin(x)\left(2^{n-1}\cos^{n-1}(x)-\frac{n-2}{1!}2^{n-3}\cos^{n-3}(x)+\frac{(n-2)(n-3)2^{n-5}\cos^{n-5}(x)}{2!}+\cdots\right)$$ I recognise the powers increased by odd terms, so I have summarised the equation up to $$\sin(x)\left(\sum_{m=0}^{\infty}\frac{2^{n-(2m+1)}\cos^{n-(2m+1)}(x)\prod_{i=2}^{m+2}(n-i)}{m!}\right)$$ However, the product $$\prod_{i=2}^{m+2}(n-i)$$ starts at $n-2$ but the initial value should begin at $1$, therefore, what is the best approach for product notation, or alternative to simplyfing the above equation?

Answer 1

I believe what you want is $\frac{(n-2)!}{(n-m-2)!}$. When $m=0$ this evaluates to $1$. When $m=1$ this evaluates to $\frac{(n-2)!}{(n-3)!}=n-2$. etc. When you combine this with the already present $m!$ in the denominator, you have the binomial coefficient $\binom{n-2}{m}$

However, I will say your formula does not look right. What you are trying to calculate is the chebyshev polynomial of the second kind, which has explicit formula listed wolfram mathworld