Generalised Hypergeometric Series

This entry provides a formalisation of the generalized hypergeometric series ${}_p F_{q}$, both as a formal power series and as a function in a Banach space. It is defined, for parameters $\textbf{a}=(a_1,\ldots,a_p)$ and $\textbf{b}=(b_1,\ldots,b_q)$ with $b_i\notin\mathbb{Z}_{{\leq}0}$ as the exponential power series \[F(\mathbf{a}; \mathbf{b}; z) = \sum_{n\geq 0} \frac{{a_1}^{\overline{n}} \cdots {a_p}^{\overline{n}}} {{b_1}^{\overline{n}} \cdots {b_q}^{\overline{n}}}\kern2pt \frac{z^n}{n!}\] where ${a}^{\overline{n}} = a(a+1)\ldots(a+n-1)$ is the Pochhammer symbol.

Basic properties of ${}_p F_{q}$ are proven (uniform convergence, continuity, holomorphicity), as well as some important properties for specific instances, such as:

• representations of various trigonometric and hyperbolic functions in terms of ${}_0 F_{1}$ and ${}_2 F_{1}$
• the contiguous identities for ${}_2 F_{1}$ and ${}_1 F_{1}$
• the transformation identity ${}_1 F_1(a;b;z) = e^z {}_1 F_1(b-a;b;-z)$ for Kummer's confluent hypergeometric function
• the fact that ${}_1 F_1$ is a solution of the ODE $a W(x) - (b - x) W'(x) - x W''(x) = 0$
• the fact that the error function can be expressed in terms of ${}_1 F_1$ as $\text{erf}(z) = \frac{2}{\sqrt{\pi}} z {}_1 F_{1}(\frac{1}{2};\frac{3}{2};-z^2)$

A regularised variant of ${}_p F_{q}$ that is also defined if $b_i\in\mathbb{Z}_{{\leq}\,0}$ is also provided.