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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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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(α,β;γ;z)=1+αβγ1z+α(α+1)β(β+1)γ(γ+1)12z2+α(α+1)(α+2)β(β+1)(β+2)γ(γ+1)(γ+2)123z3+

si1_e

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. 

limγnF(α,β;γ;z)Γ(γ)=α(α+1)(α+n)β(β+1)(β+n)(n+1)!×zn+1F(α+n+1,β+n+1;n+2;z)

si2_e   ...

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