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:

${\displaystyle {\int }_0^x}{e}^{-t}{t}^a𝑑t$ $=$ $-{\displaystyle {\int }_0^x}{t}^a𝑑e^{-t}$

$=$ $a{\displaystyle {\int }_0^x}{t}^{a-1}𝑑e^{-t}-x^a{e}^{-x}$

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

$P(a+1,x)$ $=$ $P(a,x)-{\displaystyle \frac{x^a{e}^{-x}}{\mathrm{\Gamma }(a+1)}}$

${\gamma }^{*}(a-1,x)$ $=$ $x{\gamma }^{*}(a,x)+{\displaystyle \frac{e^{-x}}{\mathrm{\Gamma }(a)}}$

Title	incomplete gamma function recurrence formula
Canonical name	IncompleteGammaFunctionRecurrenceFormula
Date of creation	2013-03-22 15:36:50
Last modified on	2013-03-22 15:36:50
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 30D30
Classification	msc 33B15