Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.6 PROPERTIES OF BINOMIAL COEFFICIENTS

We start byassumingthat (1.49) is true for some positive integern=N.Wenow

proceed to show that this implies that it must also be true forn=N+1, as follows:

(x+y)N+1=(x+y)

∑N

k=0

NC

kx

N−kyk

=

∑N

k=0

NC

kx

N+1−kyk+

∑N

k=0

NC

kx

N−kyk+1

=

∑N

k=0

NC

kx

N+1−kyk+

N∑+1

j=1

NC

j− 1 x

(N+1)−jyj,

where in the first line we have used the assumption and in the third line have

moved the second summation index by unity, by writingk+1 =j.Wenow

separate off the first term of the first sum,NC 0 xN+1, and write it asN+1C 0 xN+1;

we can do this since, as noted in (i) following (1.50),nC 0 =1foreveryn. Similarly,

the last term of the second summation can be replaced byN+1CN+1yN+1.

The remaining terms of each of the two summations are now written together,

with the summation index denoted bykin both terms. Thus

(x+y)N+1=N+1C 0 xN+1+

∑N

k=1

(N

Ck+NCk− 1

)

x(N+1)−kyk+N+1CN+1yN+1

=N+1C 0 xN+1+

∑N

k=1

N+1C

kx

(N+1)−kyk+N+1C

N+1y

N+1

=

N∑+1

k=0

N+1C

kx

(N+1)−kyk.

In going from the first to the second line we have used result (1.51). Now we

observe that the final overall equation is just the original assumed result (1.49)

but withn=N+ 1. Thus it has been shown that if the binomial expansion is

assumedto be true forn=N,thenitcanbeprovedto be true forn=N+1. But

it holds trivially forn= 1, and therefore forn= 2 also. By the same token it is

valid forn=3, 4 ,..., and hence is established for all positive integersn.

1.6 Properties of binomial coefficients

1.6.1 Identities involving binomial coefficients

There are many identities involving the binomial coefficients that can be derived

directly from their definition, and yet more that follow from their appearance in

the binomial expansion. Only the most elementary ones, given earlier, are worth

committing to memory but, as illustrations, we now derive two results involving

sums of binomial coefficients.