Determinants and Their Applications in Mathematical Physics

A.1 Miscellaneous Functions 305

δi,odd=

{

1 ,iodd,

0 ,ieven.

δi 1 i 2 ;j 1 j 2 =

{

1 , (j 1 ,j 2 )=(i 1 ,i 2 )

0 , otherwise.

The Binomial Coefficient and Gamma Function

(

n

r

)

=

{

n!

r!(n−r)!

, 0 ≤r≤n

0 , otherwise.

(

n

n−r

)

=

(

n

r

)

,

(

n

r

)

=

(

n− 1

r

)

+

(

n− 1

r− 1

)

.

The lower or upper limitr=i(→j) in a sum denotes that the limit

was originallyi, butican be replaced byjwithout affecting the sum since

the additional or rejected terms are all zero. For example,

∞

∑

r=0(→n)

ar

(r−n)!

denotes that

∞

∑

r=0

ar

(r−n)!

can be replaced by

∞

∑

r=n

ar

(r−n)!

;

n(→∞)

∑

r=0

(

n

r

)

ardenotes that

n

∑

r=0

(

n

r

)

arcan be replaced by

∞

∑

r=0

(

n

r

)

ar.

This notation has applications in simplifying multiple sums by changing

the order of summation. For example,

q

∑

n=0

n

∑

p=0

(

n

p

)

ap=

q

∑

p=0

(

q+1

p+1

)

ap.

Proof. Denote the sum on the left bySqand apply the well-known

identity

q

∑

n=p

(

n

p

)

=

(

q+1

p+1

)

Sq=

q

∑

n=0

n(→∞)

∑

p=0

(

n

p

)

ap=

∞

∑

p=0

ap

q

∑

n=0(→p)

(

n

p

)

=

∞(→q)

∑

p=0

ap

(

q+1

p+1

)

.

The result follows.