The problem provides the definition of the generating function $W^{(s)}(x, t)$ for the higher-order Gauss Fibonacci polynomials. It then attempts to derive an expression for $W^{(s)}(x, t)$ in terms of $\alpha^{(s)}(x)$ and $\beta^{(s)}(x)$, and states, after some calculations, the final expression for $W^{(s)}(x, t)$.

1. Problem Description

The problem provides the definition of the generating function $W^{(s)}(x, t)$ for the higher-order Gauss Fibonacci polynomials. It then attempts to derive an expression for $W^{(s)}(x, t)$ in terms of $\alpha^{(s)}(x)$ and $\beta^{(s)}(x)$, and states, after some calculations, the final expression for $W^{(s)}(x, t)$.

2. Solution Steps

We are given that

$W^{(s)}(x, t) = \sum_{n=0}^{\infty} GF_n^{(s)}(x)t^n$

and

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

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

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

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

After some calculations (which are not shown), the expression simplifies to

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

3. Final Answer

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