# incomplete gamma function recurrence formula

The incomplete gamma function satisfies the following recurrence formula:

 $\gamma(a+1,x)=a\gamma(a,x)-x^{a}e^{-x}$

This can be derived by integration by parts:

 $\displaystyle\int_{0}^{x}e^{-t}t^{a}\,dt$ $\displaystyle=$ $\displaystyle-\int_{0}^{x}t^{a}\,de^{-t}$ $\displaystyle=$ $\displaystyle a\int_{0}^{x}t^{a-1}\,de^{-t}-x^{a}e^{-x}$

In terms of other variants of the incomplete gamma function, the recursion relation appears as follows:

 $\displaystyle P(a+1,x)$ $\displaystyle=$ $\displaystyle P(a,x)-{x^{a}e^{-x}\over\Gamma(a+1)}$ $\displaystyle\gamma^{*}(a-1,x)$ $\displaystyle=$ $\displaystyle x\gamma^{*}(a,x)+{e^{-x}\over\Gamma(a)}$
Title incomplete gamma function recurrence formula IncompleteGammaFunctionRecurrenceFormula 2013-03-22 15:36:50 2013-03-22 15:36:50 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Theorem msc 30D30 msc 33B15