2f1

2F1 refers to the Gauss hypergeometric function, commonly written as _2F1(a,b;c;z). It is a classical special function that arises in many areas of mathematics and its applications. The function is defined by a power series in z:

_2F1(a,b;c;z) = sum_{n=0}^\infty (a)_n (b)_n / (c)_n · z^n / n!,

where (q)_n is the Pochhammer symbol (rising factorial). The series converges for |z| < 1 and defines an analytic function there; it can be extended to larger domains through analytic continuation.

The Gauss hypergeometric function satisfies the hypergeometric differential equation

z(1 - z) y'' + [c - (a + b + 1) z] y' - a b y = 0.

It exhibits symmetry in its first two parameters, demonstrating that _2F1(a,b;c;z) = _2F1(b,a;c;z). An Euler-type integral representation also exists:

_2F1(a,b;c;z) = Γ(c) / [Γ(b) Γ(c - b)] ∫_0^1 t^{b-1} (1 - t)^{c - b - 1} (1 - z t)^{-a} dt,

valid for Re(c) > Re(b) > 0. Transformations relate values at z to 1 - z or 1/z, such as Euler’s transformation: _2F1(a,b;c;z) = (1 - z)^{c - a - b} _2F1(c - a, c - b; c; z).

Special cases occur when a or b is a nonpositive integer, yielding polynomials related to Jacobi and Gegenbauer families. For certain parameter choices, _2F1 reduces to elementary functions.

Applications span probability (beta-binomial distributions), mathematical physics (quantum mechanics, Feynman integrals), and geometric problems. It also serves as a fundamental building block in the study of differential equations and in the representation theory of Lie groups.