hypergeometric function
/ the symbol is 2-F-1 /
Many of the named functions so far — geometric series, the exponential, the logarithm, Legendre and Chebyshev polynomials, the elliptic integrals — look unrelated. The hypergeometric function reveals them as special cases of one master series. It is a single parametrised function from which a remarkable fraction of the functions of mathematical physics drop out by choosing its parameters.
The Gauss hypergeometric function, written 2F1(a, b; c; x), is the power series 1 plus (a b over c) x over 1! plus (a(a+1) b(b+1) over c(c+1)) x^2 over 2! plus ... — a power series whose successive coefficient ratios are a simple rational function of the index. The plain geometric series 1 plus x plus x^2 plus ... is the special case 2F1(1, 1; 1; x); the logarithm, arcsine, the complete elliptic integrals and the classical orthogonal polynomials are all 2F1 (or its confluent limit 1F1) with particular parameters. The series converges for |x| less than 1 and is continued beyond by integral representations and functional relations. Letting two parameters merge produces the confluent hypergeometric function 1F1, which captures the Bessel, Hermite and Laguerre families.
The hypergeometric function is the great unifier: it is the general solution of the hypergeometric differential equation, a second-order linear ODE with exactly three regular singular points, and almost every classical special function is a special or limiting case. In applied work it gives closed-form answers in probability (it sums many discrete distributions), in the analytic solution of physical ODEs, and in evaluating families of integrals. Knowing a result is hypergeometric instantly connects it to a vast library of identities and fast numerical methods.
Set a = b = 1 and c = 1: every coefficient collapses to 1 and 2F1(1, 1; 1; x) = 1 plus x plus x^2 plus ... = 1 over (1 minus x), the ordinary geometric series — the simplest special case of the master function.
Even the humble geometric series is a single point in the hypergeometric family's parameter space.
The labels 2 and 1 in 2F1 count the parameters (two on top, one below), not the degree; the series terminates into a polynomial only when a or b is a negative integer, which is how it produces the classical orthogonal polynomials.