Hypergeometric functions often refer to a family of functions ${}_p F_q$ represented by a corresponding series, where $p,q$ are non-negative integers. The case ${}_2F_1(a,b;c;z)$ is a special case of particular importance; it is known as the Gaussian, or ordinary, hypergeometric function.
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813). It refers to a family of functions $$ _p F_q(z) = F(\{a\};\{b\};z) =\sum_{n=0}^{\infty} \frac{(a_1)_n\cdots (a_p)_n}{n! (b_1)_n\cdots (b_q)_n} z^n $$Here $(a)_n=a(a+1)\cdots(a+n-1)$ is the increasing Pochhammer symbol; care should be taken, as the notation is occasionally ambiguous. For $p=q+1$ the series converges for $|z|\le 1$; for $p\le q$ it converges for any complex $z$.
The case ${}_2F_1(z)$ is particularly important. Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterization by Bernhard Riemann of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for ${}_2F_1(z)$, examined in the complex plane, could be characterized (on the Riemann sphere) by its three regular singularities. Many special and elementary functions are specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).
Evaluation of higher hypergeometric functions is a complex and interesting topic. Important formulas include Clausen's formula: $$ \left({_2F_1}(\{a,b\};\{a+b+1/2\};z\right)^2 = {_3F_2}(\{2a,2b,a+b\};\{2a+2b,a+b+1/2\};z); $$the positivity of the RHS for real $z$ is useful. Generalizations include using $q$-binomial coefficients, the Meijer-G function, and Fox-Wright functions, where the Pochhammer symbols are taken as gamma functions.
Tags containing this question often involve special-functions, closed-form, zeta-functions.
