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Special Families of Polynomials 3
number x, is defined as
Γ(x) :=
+∞
0
t x−1 e −t dt.
Proving the following functional equation is straightforward after integration by parts:
(1.2.1) Γ(x + 1) = xΓ(x) , x > 0.
In particular, when n is an integer, we have a fundamental relation obtained from (1.2.1)
by induction, i.e. Γ(n + 1) = n!.
Another important relation is
(1.2.2) Γ(x) Γ(1 − x) = π
, 0 < x < 1.
sinπ x
From (1.2.2) one can easily obtain Γ √ 1
2 = π. A useful equality, which can be deduced
from the definition of the Beta function (see for instance luke (1969), Vol.1), is
(1.2.3)
1
−1
(1 + t) x−1 (1 − t) y−1 x+y−1 Γ(x) Γ(y)
dt = 2 , x > 0, y > 0.
Γ(x + y)
The binomial coefficients are generalized according to
(1.2.4)
x
:=
k
Γ(x + 1)
, k ∈ N, x > k − 1.
k! Γ(x − k + 1)
The formulas considered here, as well as many other properties of the Gamma
function are widely discussed in luke(1969), Vol.1; srivastava and manocha(1984),
hochstadt(1971).
The numerical evaluation of the Gamma function is generally performed by reduc-
ing the computation shifting backwards to the interval ]0,1] using (1.2.1). The interval
can be further reduced to ] 1
2 , 1] by taking into account (1.2.2). Finally, suitable series
expansions centered about 1 can be implemented. For other theoretical and compu-
tational aspects we can refer for instance to luke(1969), Vol.2. We only mention the
following expansion: