Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 31/Hypergeometric Functions
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Hypergeometric Functions
- A solution of $31.1$ is given by
$31.2$: Gaussian Hypergeometric Function
The Gaussian hypergeometric function is an instance of a generalized hypergeometric function, given for $\size z < 1$ by:
\(\ds \map F {a, b; c; z}\) | \(:=\) | \(\ds \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} } \dfrac {z^n} {n!}\) | where $x^{\overline n}$ denotes the $n$th rising factorial power of $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \dfrac {a b} {1! \, c} z + \dfrac {a \paren {a + 1} b \paren {b + 1} } {2! \, c \paren {c + 1} } z^2 + \dfrac {a \paren {a + 1} \paren {a + 2} b \paren {b + 1} \paren {b + 2} } {3! \, c \paren {c + 1} \paren {c + 2} } z^3 + \cdots\) |
- If $a$, $b$, $c$ are real, then the series converges for $-1 < x < 1$ provided that $c - \paren {a + b} > 1$.