One hypergeometric function quietly contains logarithms, arcsines and Bessel functions
Mathematicians have published many thousands of identities for the hypergeometric function, yet nobody knows an algorithm that could generate them all, or even a scheme for organising them. That richness comes from its reach: logarithms, inverse sines, Legendre functions and whole families of polynomials all turn out to be special cases hiding inside it.
The name reflects its ancestry. Take the ordinary geometric series one plus z plus z squared and so on, then let each coefficient become a ratio built from rising products of three parameters, usually called a, b and c. The result, for values of z inside the unit circle, is Gauss's hypergeometric function. Choose the parameters suitably and it collapses back to the plain geometric series, so it is best seen as a grand generalisation of it. If a or b is a negative whole number the series stops early and becomes a polynomial; if c is zero or a negative integer it blows up.
John Wallis coined the phrase hypergeometric series in his 1655 book Arithmetica Infinitorum. Leonhard Euler studied such series, but Carl Friedrich Gauss produced the first full systematic treatment in 1813. Ernst Kummer followed in 1836, and in 1857 Bernhard Riemann gave a fundamental characterisation: the function is pinned down by the second-order differential equation it solves, which on the Riemann sphere has exactly three regular singular points. Hermann Schwarz later listed the cases where its solutions are algebraic.
That differential equation is the key to its ubiquity. Any second-order linear equation with three regular singular points can be transformed into it, which is why Legendre functions can be written in hypergeometric terms in many ways. Jacobi polynomials and their relatives, including the Legendre, Chebyshev, Gegenbauer and Zernike polynomials, are special cases too. A limiting version, Kummer's confluent function, captures Bessel functions and most of the everyday functions of mathematical physics.
Outside the unit circle the function can be extended along paths that avoid the troublesome points at one and infinity, and software usually places a branch cut along the real line beyond one. Its derivative is simply another hypergeometric function with each parameter nudged up by one. The standard catalogues of identities by Erdélyi and colleagues and by Olde Daalhuis are still the standard places to look, and the search for automatic ways to discover new identities is still active research.
Source: Hypergeometric function
