Hypergeometric pFq

Let $p,q \in \{0,1,2,\ldots\}$ and $a_j,b_{\ell} \in \mathbb{R}$ for $j=1,\ldots,p$ and $\ell=1,\ldots,q$. Define the generalized hypergeometric function $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$

Convergence

If any of the $a_j$'s is a a nonpositive integer, then the series terminates and is a polynomial.

If any of the $b_{\ell}$'s is a nonpositive integer, the series diverges because of divison by zero.

The remaining convergence of the series can be split into three cases:

Case I: $p<q+1$

Proposition: The series ${}_pF_q$ converges for all $t \in \mathbb{C}$.
Proof:

Case II: $p=q+1$

Proposition: The series ${}_pF_q$ converges for all $t\in \mathbb{C}$ with $|t|<1$.
Proof:

Case III: $p>q+1$

Proposition: The series ${}_pF_q$ diverges for all $t \in \mathbb{C}$.
Proof: