Table of Integrals, Series, and Products, 8th Edition

9

Special Functions

9.1 Hypergeometric Functions

9.10 Definition

9.100 A hypergeometric series is a series of the form

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

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. 

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

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