Gruber P. Convex and Discrete Geometry

(1) g(n + 1) = n! for n = 0, 1,...
1 Convex Functions of One Variable 19
The main step of the proof is to show that Properties (i)–(iii) yield the formula
n
(2) g(x) = lim
n→∞
xn! for x > 0.
x(x + 1) ···(x + n)
Assume first that 0 < x ≤ 1. The logarithmic convexity (iii), together with the
functional equation (ii) and (1), shows that
(3) g(n + 1 + x) = g � (1 − x)(n + 1) + x(n + 2) �
≤ g(n + 1) 1−x g(n + 2) x = (n + 1) x n!,
(4) n! =g(n + 1) = g � x(n + x) + (1 − x)(n + 1 + x) �
≤ g(n + x) x g(n + 1 + x) 1−x = (n + x) −x g(n + 1 + x) x g(n + 1 + x) 1−x
= (n + x) −x g(n + 1 + x).
An immediate consequence of the functional equation (ii) is the identity
(5) g(n + 1 + x) = (n + x)(n − 1 + x) ···xg(x).
Combining (3)–(5), we obtain the following inequalities:
�
1 + x
n
� x
≤
(n + x)(n − 1 + x) ···xg(x)
n x n!
�
≤ 1 + 1
n
� x
for n = 1, 2,...,
which, in turn, yields (2) in case 0 < x ≤ 1.
Assume, second, that x > 1. Using the functional equation (ii), this can be reduced
to the case already settled: choose an integer m such that 0 < x − m ≤ 1.
Then
g(x) = (x − 1) ···(x − m)g(x − m)
n
= (x − 1) ···(x − m) lim
n→∞
x−m n!
(x − m) ···(x − m + n)
�
n
= lim
n→∞
xn! (x + n) ···(x + n − m + 1)
·
x(x + 1) ···(x + n) nm �
n
= lim
n→∞
xn! x(x + 1) ···(x + n)
by (ii) and the already settled case of (2). Thus (2) also holds for x > 1, which
concludes the proof of (2).
Since Ɣ has Properties (i)–(iii) and (2) was proved using only these properties,
we see that
n
(6) Ɣ(x) = g(x) = lim
n→∞
xn! for x > 0. ⊓⊔
x(x + 1) ···(x + n)
Formula (6) was used by Euler in 1729 to introduce the gamma function. It is
sometimes named after Gauss. Theorem 1.11 can be used to derive other properties
of the gamma function. See, e.g. Webster [1016].