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9

# Special Functions

## 9.1 Hypergeometric Functions

### 9.10 Definition

9.100 A hypergeometric series is a series of the form

$F\left(\alpha ,\beta ;\gamma ;z\right)=1+\frac{\alpha \cdot \beta }{\gamma \cdot 1}z+\frac{\alpha \left(\alpha +1\right)\beta \left(\beta +1\right)}{\gamma \left(\gamma +1\right)\cdot 1\cdot 2}{z}^{2}+\frac{\alpha \left(\alpha +1\right)\left(\alpha +2\right)\beta \left(\beta +1\right)\left(\beta +2\right)}{\gamma \left(\gamma +1\right)\left(\gamma +2\right)\cdot 1\cdot 2\cdot 3}{z}^{3}+\dots$

9.101 A hypergeometric series terminates if α or β is equal to a negative integer or to zero. For γ = −n (n = 0, 1, 2, ), the hypergeometric series is indeterminate if neither α nor β is equal to –m (where m < n and m is a natural number). However,

1.

$\underset{\gamma \to -n}{\mathrm{lim}}\frac{F\left(\alpha ,\beta ;\gamma ;z\right)}{\mathrm{\Gamma }\left(\gamma \right)}=\frac{\alpha \left(\alpha +1\right)\dots \left(\alpha +n\right)\beta \left(\beta +1\right)\dots \left(\beta +n\right)}{\left(n+1\right)!}×{z}^{n+1}F\left(\alpha +n+1,\beta +n+1;n+2;z\right)$

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