Mathematical Methods for Physics and Engineering - Matematica.NET

1.6 PROPERTIES OF BINOMIAL COEFFICIENTSWe have specifically included the second equality to emphasise the symmetricalnature of the relationship with respect to p and q.Further identities involving the coefficients can be obtained by giving x and yspecial values in the defining equation (1.49) for the expansion. If both are setequal to unity then we obtain (using the alternative notation so as to producefamiliarity with it)( ( ( ( n n n n+ + + ···+ =20)1)2)n)n , (1.55)whilst setting x = 1 and y = −1 yields( ( ( ( )n n n n− + − ···+(−1)0)1)2)n n=0. (1.56)1.6.2 Negative and non-integral values of nUp till now we have restricted n in the binomial expansion to be a positiveinteger. Negative values can be accommodated, but only at the cost of an infiniteseries of terms rather than the finite one represented by (1.49). For reasons thatare intuitively sensible and will be discussed in more detail in chapter 4, veryoften we require an expansion in which, at least ultimately, successive terms inthe infinite series decrease in magnitude. For this reason, if x>ywe consider(x + y) −m ,wherem itself is a positive integer, in the form(x + y) n =(x + y) −m = x −m ( 1+ y x) −m.Since the ratio y/x is less than unity, terms containing higher powers of it will besmall in magnitude, whilst raising the unit term to any power will not affect itsmagnitude. If y>xthe roles of the two must be interchanged.We can now state, but will not explicitly prove, the form of the binomialexpansion appropriate to negative values of n (n equal to −m):∑∞ ( y) k(x + y) n =(x + y) −m = x −m −m C k , (1.57)xwhere the hitherto undefined quantity −m C k , which appears to involve factorialsof negative numbers, is given by−m k m(m +1)···(m + k − 1) k (m + k − 1)!C k =(−1) =(−1) =(−1) k m+k−1 C k .k!(m − 1)!k!(1.58)The binomial coefficient on the extreme right of this equation has its normalmeaning and is well defined since m + k − 1 ≥ k.Thus we have a definition of binomial coefficients for negative integer valuesof n in terms of those for positive n. The connection between the two may not29k=0