Generating function of trigonometric fuction

Question

Find exponential generating function for following sequence:

$s_{n} = \sin{nt}$

the answer should be in terms of trigonometric functions. The exponential generating function is defined as:

$S(x)= \sum_{n\geq 0} s_{n} \frac{x^n}{n!} $

Answer 1

You can determine a function that simultaneously "generates" both $sin(nt)$ and $cos(nt)$. If you assume $x$ to be real, then you're looking for the imaginary part of the coefficient of $\frac{x^n}{n!}$ in the following expression.

$$ \sum_{n=0}^\infty \left(cos(nt) + i.sin(nt)\right)\frac{x^n}{n!} \\ = \sum_{n=0}^\infty e^{int}\frac{x^n}{n!} \\ = \sum_{n=0}^\infty \frac{e^{n(ln(x) + it)}}{n!} = e^{e^{ln(x) + it}} \\ = e^{x.e^{it}} $$ Is this what you are looking for?