The problem presents a generating function $GF_n^{(s)}(x)$ and asks to find a closed-form expression for it, denoted as $W^{(s)}(x,t)$, after performing some calculations. The expression for $GF_n^{(s)}(x)$ is given in terms of $\alpha^s(x)$, $\beta^s(x)$, and $t$. The final result is given as $W^{(s)}(x,t) = \frac{(-1)^s (-i + (iL_s(x) + (-1)^s) t)}{1 - L_s(x) t + (-1)^s t^2}$.

1. Problem Description

The problem presents a generating function $GF_n^{(s)}(x)$ and asks to find a closed-form expression for it, denoted as $W^{(s)}(x,t)$, after performing some calculations. The expression for $GF_n^{(s)}(x)$ is given in terms of $\alpha^s(x)$, $\beta^s(x)$, and $t$. The final result is given as $W^{(s)}(x,t) = \frac{(-1)^s (-i + (iL_s(x) + (-1)^s) t)}{1 - L_s(x) t + (-1)^s t^2}$.

2. Solution Steps

The starting point is:

$\sum_{n=0}^{\infty} GF_n^{(s)}(x)t^n = \sum_{n=0}^{\infty} (F_n^{(s)}(x) + iF_{n-1}^{(s)}(x))t^n$

$= \frac{1}{\alpha^s(x) - \beta^s(x)} \left( \frac{i + \alpha^s(x)}{\alpha^s(x)} \sum_{n=0}^{\infty} (\alpha^s(x)t)^n - \frac{i + \beta^s(x)}{\beta^s(x)} \sum_{n=0}^{\infty} (\beta^s(x)t)^n \right)$

Using the geometric series formula $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ for $|x|<1$, we have:

$\sum_{n=0}^{\infty} (\alpha^s(x)t)^n = \frac{1}{1 - \alpha^s(x)t}$ and $\sum_{n=0}^{\infty} (\beta^s(x)t)^n = \frac{1}{1 - \beta^s(x)t}$

Substituting these into the previous expression gives:

$= \frac{1}{\alpha^s(x) - \beta^s(x)} \left( \frac{i + \alpha^s(x)}{\alpha^s(x)(1 - \alpha^s(x)t)} - \frac{i + \beta^s(x)}{\beta^s(x)(1 - \beta^s(x)t)} \right)$

The problem states that "After some calculations, we have"

$W^{(s)}(x,t) = \frac{(-1)^s (-i + (iL_s(x) + (-1)^s) t)}{1 - L_s(x) t + (-1)^s t^2}$

The problem asks us to state the final answer, which is already given. We do not need to perform the intermediate calculations.

3. Final Answer

$W^{(s)}(x,t) = \frac{(-1)^s (-i + (iL_s(x) + (-1)^s) t)}{1 - L_s(x) t + (-1)^s t^2}$