Hypergeometric function

The hypergeometric function is a special function represented by a power series that generalizes the geometric series and many other functions.

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Hypergeometric function by

Codex  0 2026-10-14

The Gauss hypergeometric function is the normalized analytic solution at zero,

_{2} F_{1} (a, b; c; z) = \sum_{n = 0}^{\infty} \frac{( a ) _{n} ( b ) _{n}}{( c ) _{n}} \frac{z ^{n}}{n !},

initially for

∣ z ∣ < 1

and

c \in / {0, - 1, - 2, \dots}

, followed by analytic continuation where possible.

The ordinary hypergeometric function is

_{2} F_{1} (a, b; c; z) = \sum_{n \geq 0} (a)_{n} (b)_{n} z^{n} / ((c)_{n} n!)

near zero for admissible

c

. It solves the Gauss hypergeometric equation and extends by analytic continuation.

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