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
Derivatives of Hypergeometric pFq
Differential equation for Hypergeometric pFq

Examples

${}_0F_0$

Exponential in terms of hypergeometric 0F0

${}_0F_1$

Relationship between cosine and hypergeometric 0F1
Relationship between sine and hypergeometric 0F1
Relationship between cosh and hypergeometric 0F1
Relationship between sinh and hypergeometric 0F1
Relationship between Bessel J sub nu and hypergeometric 0F1

${}_1F_2$

Relationship between Struve function and hypergeometric pFq

${}_2F_0$

Bessel polynomial generalized hypergeometric

${}_2F_1$

z2F1(1,1;2,-z) equals log(1+z)
Relationship between arcsin and hypergeometric 2F1
Relationship between arctan and hypergeometric 2F1
Relationship between Chebyshev T and hypergeometric 2F1
Relationship between Chebyshev U and hypergeometric 2F1
Relationship between Legendre polynomial and hypergeometric 2F1
Relationship between incomplete beta and hypergeometric 2F1

Videos

Special functions - Hypergeometric series

References

Notes on hypergeometric functions
Rainville's Special Functions
Abramowitz and Stegun
Note on a hypergeometric series - Cayley