Page 320 - Determinants and Their Applications in Mathematical Physics

A.1 Miscellaneous Functions 305

1,i odd,
δ i,odd =
0,i even.

1, (j 1 ,j 2 )=(i 1 ,i 2 )
=
δ i 1 i 2 ;j 1 j 2
0, otherwise.
The Binomial Coefficient and Gamma Function

n!
n , 0 ≤ r ≤ n
= r!(n−r)!
r 0, otherwise.

n n
= ,
n − r r

n n − 1 n − 1
= + .
r r r − 1
The lower or upper limit r = i (→ j) in a sum denotes that the limit
was originally i, but i can be replaced by j without affecting the sum since
the additional or rejected terms are all zero. For example,
∞ ∞ ∞

denotes that a r can be replaced by a r ;
a r
(r − n)! (r − n)! (r − n)!
r=0(→n) r=0 r=n
n(→∞)
n
∞
n n n

a r denotes that a r can be replaced by a r .
r r r
r=0 r=0 r=0
This notation has applications in simplifying multiple sums by changing
the order of summation. For example,
n
q
n q +1
q

a p = a p .
p p +1
n=0 p=0 p=0
Proof. Denote the sum on the left by S q and apply the well-known
identity
q
n q +1

=
p p +1
n=p
n(→∞)
q ∞ q
n n
S q = a p =
p a p p
n=0 p=0 p=0 n=0(→p)
∞(→q)
q +1
= .
p +1
a p
p=0
The result follows.