I’ve asserted numerous times here that hypergeometric functions come up very often in applied math, but I haven’t said why. This post will give one reason why.
One way to classify functions is in terms of the differential equations they satisfy. Elementary functions satisfy simple differential equations. For example, the exponential function satisfies
y′ = y
and sine and cosine satisfy
y″ + y = 0.
Second order linear differential equations are common and important because Newton’s are second order linear differential equations. For the rest of the post, differential equations will be assumed to be second order and linear, with coefficients that are analytic except possibly at isolated singularities.
ODEs can be classified in terms of the sigularities of their coefficients. The equations above have no singularities and so their solutions are some of the most fundamental equations.
If an ODE has two or fewer regular singular points , it can be solved in terms of elementary functions. The next step in complexity is to consider differential equations whose coefficients have three regular singular points. Here’s the kicker:
Every ODE with three regular singular points can be transformed into the hypergeometric ODE.
So one way to think of the hypergeometric differential equation is that it is the standard ODE with three regular singular points. These singularities are at 0, 1, and ∞. If a differential equation has singularities elsewhere, a change of variables can move the singularities to these locations.
The hypergeometric differential equation is
x(x – 1) y″ + (c – (a + b + 1)x) y′ + ab y = 0.
The hypergeometric function F(a, b; c; x) satisfies this equation, and if c is not an integer, then a second independent solution to the equation is
x1-c F(1 + a – c, 1 + b – c; 2 – c; x).
To learn more about the topic of this post, see Second Order Differential Equations: Special Functions and Their Classification by Gerhard Kristensson.
 This includes singular points at infinity. A differential equation is said to have a singularity at infinity if under the change of variables t = 1/x the equation in u has a singularity at 0. For example, Airy’s equation
y″ + x y = 0
has no singularities in the finite complex plane, and it has no solution in terms of elementary functions. This does not contradict the statement above because the equation has a singularity at infinity.