Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.6 PROPERTIES OF BINOMIAL COEFFICIENTS

We have specifically included the second equality to emphasise the symmetrical

nature of the relationship with respect topandq.

Further identities involving the coefficients can be obtained by givingxandy

special values in the defining equation (1.49) for the expansion. If both are set

equal to unity then we obtain (using the alternative notation so as to produce

familiarity with it)
(
n
0

)

+

(

n

1

)

+

(

n

2

)

+···+

(

n

n

)

=2n, (1.55)

whilst settingx= 1 andy=−1 yields
(
n
0

)

−

(

n

1

)

+

(

n

2

)

−···+(−1)n

(

n

n

)

=0. (1.56)

1.6.2 Negative and non-integral values ofn

Up till now we have restrictednin the binomial expansion to be a positive

integer. Negative values can be accommodated, but only at the cost of an infinite

series of terms rather than the finite one represented by (1.49). For reasons that

are intuitively sensible and will be discussed in more detail in chapter 4, very

often we require an expansion in which, at least ultimately, successive terms in

the infinite series decrease in magnitude. For this reason, ifx>ywe consider

(x+y)−m,wheremitself is a positive integer, in the form

(x+y)n=(x+y)−m=x−m

(

1+

y

x

)−m

.

Since the ratioy/xis less than unity, terms containing higher powers of it will be

small in magnitude, whilst raising the unit term to any power will not affect its

magnitude. Ify>xthe roles of the two must be interchanged.

We can now state, but will not explicitly prove, the form of the binomial

expansion appropriate to negative values ofn(nequal to−m):

(x+y)n=(x+y)−m=x−m

∑∞

k=0

−mC

k

(y

x

)k

, (1.57)

where the hitherto undefined quantity−mCk, which appears to involve factorials

of negative numbers, is given by

−mC

k=(−1)

km(m+1)···(m+k−1)

k!

=(−1)k

(m+k−1)!

(m−1)!k!

=(−1)km+k−^1 Ck.

(1.58)

The binomial coefficient on the extreme right of this equation has its normal

meaning and is well defined sincem+k− 1 ≥k.

Thus we have a definition of binomial coefficients for negative integer values

ofnin terms of those for positiven. The connection between the two may not