In Concrete Mathematics, section 5.5, the book discussed degenerate cases for hypergeometric series. But slightly later on, after establishing the gamma function, $\Gamma(z+1)=z!$, $(-z)!\Gamma(z)=\frac{\pi}{sin (\pi z)}$, generalised rising and falling powers, $z^{\underline{w}}=$ and the limit definition for binomial coefficients $\binom{z}{w}=\lim_{\zeta \rightarrow z}\lim_{\omega \rightarrow w}\frac{\zeta !}{\omega !(\zeta - \omega)!}$ (to be used whenever $\frac{z!}{w!(z-w)!}$ is undefined), the book goes on to use factorial cancellation liberally without considering any degenerate cases.
Factorial canceling on expansion of binomial coefficients on Concrete Mathematics (This links to a related qn)
To calculate a hypergeometric representation of a series $\sum_{k\geq 0}t_k$, we consider the term ratio $\frac{t_{k+1}}{t_k}$, so my definition for a degenerate case would be:
Whenever $\binom{z}{w}\neq \frac{z!}{w!(z-w)!}$, i.e. both numerator and denominator $\rightarrow \infty$, in this case we need to use the limit definition.
Whenever $t_{k+1} = \infty$ and $\frac{1}{t_k} = 0$, this can occur when there is a negative factorial in both expressions.
The book goes on to transform Vandermonde's convolution into a hypergeometric. $$\begin{align}&\sum_k\binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}\\ &t_k=\frac{r!}{(r-k)!k!}\frac{s!}{(s-n+k)!(n-k)!}\\ &\frac{t_{k+1}}{t_k}=\frac{k-r}{k+1}\frac{k-n}{k+s-n+1}\\ &\binom{s}{n}F(-r,-n;s-n+1|1)=\binom{r+s}{n},\, n\in \mathbb{Z}^+ \end{align}$$ The book seems to imply that the last equality holds for any real $r,s$, which seems problematic to me. For e.g. when integer $s<n$, the hypergeometric formula isn't even defined. The book spoke previously of considering the limit $$\lim_{\epsilon \rightarrow 0}F(-r,-n;s-n+1+\epsilon|1)$$ in such cases, but in this section, the book didn't make any mention of degenerate cases. There are also values of $r,s$ that lead to degenerate situations of the 2 points I raised above.
So how should I interpret this portion of the book?