Hypergeometric pFq
Let $p,q \in \{0,1,2,\ldots\}$ and $a_j,b_{\ell} \in \mathbb{R}$ for $j=1,\ldots,p$ and $\ell=1,\ldots,q$. We will use the notation $\vec{a}=\displaystyle\prod_{j=1}^p a_j$ and $\vec{b}=\displaystyle\prod_{\ell=1}^q b_{\ell}$ and we define the notations $$\vec{a}^{\overline{k}} = \displaystyle\prod_{j=1}^p a_j^{\overline{k}},$$ and $$\vec{a}+k = \displaystyle\prod_{j=1}^p (a_j+k),$$ (and similar for $\vec{b}^{\overline{k}}$). Define the generalized hypergeometric function $${}_pF_q(a_1,a_2,\ldots,a_p;b_1,\ldots,b_q;t)={}_pF_q(\vec{a};\vec{b};t)=\displaystyle\sum_{k=0}^{\infty}\dfrac{\displaystyle\prod_{j=1}^p a_j^{\overline{k}}}{\displaystyle\prod_{\ell=1}^q b_{\ell}^{\overline{k}}} \dfrac{t^k}{k!}.$$
Properties
Convergence of Hypergeometric pFq
Hypergeometric pFq terminates to a polynomial if an a_j is a nonpositive integer
Hypergeometric pFq diverges if a b_j is a nonpositive integer
Hypergeometric pFq converges for all z if p less than q+1
Hypergeometric pFq converges in the unit disk if p=q+1
Hypergeometric pFq diverges if p greater than q+1
Derivatives of Hypergeometric pFq
Differential equation for Hypergeometric pFq
Particular hypergeometric functions
Hypergeometric 0F0
Hypergeometric 1F0
Hypergeometric 0F1
Hypergeometric 1F1
Hypergeometric 2F1
Hypergeometric 1F2
Hypergeometric 2F0
Hypergeometric 2F1
Videos
Special functions - Hypergeometric series (9 March 2011)
References
Notes on hypergeometric functions
Rainville's Special Functions
Abramowitz and Stegun
Note on a hypergeometric series - Cayley
Hypergeometric ${}_pF_q$
