
Chapter 3 0
Special Functions
and Their Properties
⋆
Throughout Chapter 30 it is assumed that
n
is a positive integer unless otherwise spec-
ified.
30.1 Some Coefficients, Symbols, and Numbers
30.1.1 Bin omial Coefficients
◮ Definitions.
C
k
n
=
n
k
=
n!
k! (n −k)!
, where k = 1, . . . , n;
C
0
a
= 1, C
k
a
=
a
k
= (−1)
k
(−a)
k
k!
=
a(a − 1) . . . (a −k + 1)
k!
, where k = 1, 2, . . .
Here a is an arbitrary real number.
◮ Generalization. Some properties.
General case:
C
b
a
=
Γ(a + 1)
Γ(b + 1)Γ(a −b + 1)
, where Γ(x) is the gamma function.
Properties:
C
0
a
= 1, C
k
n
= 0 for k = −1, −2, . . . or k > n,
C
b+1
a
=
a
b + 1
C
b
a−1
=
a − b
b + 1
C
b
a
, C
b
a
+ C
b+1
a
= C
b+1
a+1
,
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