Page 274 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

260 Approximation of Integrals

For n = 1, 2,...,
1 1 1
ψ(x + n) = + +· · · + + ψ(x), x > 0.
x x + 1 x + n − 1

Incomplete gamma function
The limits of the integral defining the gamma function are 0 and ∞. The incomplete gamma
function is the function obtained by restricting the region of integration to (0, y):
y
γ (x, y) = t x−1 exp(−t) dt, x > 0, y > 0.
0
Thus, considered as a function of y for fixed x, the incomplete gamma function is, aside
from a normalization factor, the distribution function of the standard gamma distribution
with index x.
The following theorem shows that, like the gamma function, the incomplete gamma
function satisfies a recursive relationship; the proof is left as an exercise.

Theorem 9.5. For x > 0 and y > 0,
x
γ (x + 1, y) = xγ (x, y) − y exp(−y).
It is sometimes convenient to consider the function
∞

(x, y) = t x−1 exp(−t) dt = (x) − γ (x, y).
y
When x is an integer, (x, y)/ (x), and, hence, γ (x, y)/ (x), can be expressed as a
finite sum.

Theorem 9.6. For n = 1, 2,...,
n−1
(n, y) y j
= exp(−y), y > 0;
(n) j!
j=0
equivalently,
n−1 j
γ (n, y) y
= 1 − exp(−y), y > 0.
(n) j!
j=0
Proof. Note that, using the change-of-variable s = t − y,
∞ ∞

t n−1 exp(−t) dt = exp(−y) (s + y) n−1 exp(−s) ds
y 0
∞ n−1

n − 1 n−1− j j
= exp(−y) s y exp(−s) ds
0 j=0 j
n−1
j n − 1
= exp(−y)y (n − j)
j
j=0
n−1 j
y
= exp(−y) (n − 1)!.
j!
j=0