Page 275 - Elements of Distribution Theory

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

9.2 Some Useful Functions 261

That is,

n−1 j
y
(n, y) = exp(−y) (n),
j!
j=0
proving the result.

For general values of x, the following result gives two series expansions for γ (x, y).

Theorem 9.7. For x > 0 and y > 0,

∞ j x+ j
(−1) y
γ (x, y) = (9.1)
j! x + j
j=0
∞
(x) x+ j
= exp(−y) y . (9.2)
(x + j + 1)
j=0
Proof. Recall that, for all x,
∞
j
exp(x) = x /j!;
j=0
hence,

∞
y y
j j
γ (x, y) = t x−1 exp(−t) dt = t x−1 (−1) t /j! dt
0 0 j=0
∞ j y
(−1) x+ j−1
= t dt
j!
j=0 0
∞ j x+ j
(−1) y
= .
j! x + j
j=0
Note that the interchanging of summation and integration is justified by the fact that
∞ y ∞ x+ j
1 1 y
x+ j−1 x
t dt = ≤ y exp(y) < ∞
j! j! x + j
j=0 0 j=0
for all y, x. This proves (9.1).
Now consider (9.2). Using the change-of-variable u = 1 − t/y,we may write
y 1
x
γ (x, y) = t x−1 exp(−t) dt = y (1 − u) x−1 exp{−y(1 − u)} du
0 0
1
x
= y exp(−y) (1 − u) x−1 exp(yu) du
0
1 ∞
j
x−1
x
= y exp(−y) (1 − u) (yu) /j! du.
0 j=0