binomial formula for negative integer powers


For negative integer powers, the binomial formulaMathworldPlanetmath can be written in terms of binomial coefficientsMathworldPlanetmath like so:

(1-x)-n=m=1(m+n-1n-1)xm

Proof:   We shall prove this by inductionMathworldPlanetmath on n. First, note that, if n=1, then (m0)=1, so our formulaMathworldPlanetmathPlanetmath reduces to

(1-x)-1=m=1xm,

which is the formula for the sum of an infinite geometric series.

Next, suppose that the formula is valid for a certain value of n. Then we have

(1-x)-n-1=(1-x)-1(1-x)-n=(k=0xk)(m=0(m+n-1n-1)xm)

The productPlanetmathPlanetmath of sums can be rewritten as the following double sum:

m=0k=0m(n+k-1n-1)xm

The easiest way to see this is by rearranging the double sum as follows and adding columns

x0m=0(m+n-1n-1)xm=(n-1n-1)+(nn-1)x+(n+1n-1)x2+(n+2n-1)x3+(n+3n-1)x4+x1m=0(m+n-1n-1)xm=(n-1n-1)x+(nn-1)x2+(n+1n-1)x3+(n+2n-1)x4+x2m=0(m+n-1n-1)xm=(n-1n-1)x2+(nn-1)x3+(n+1n-1)x4+x3m=0(m+n-1n-1)xm=(n-1n-1)x3+(nn-1)x4+............

To evaluate the finite sums, we shall use the following identity for binomial coefficients. (See the entry http://planetmath.org/node/273“binomial coefficient” for more information about this identity.)

k=0m(n+k-1n-1)=(m+nn)

Inserting this result value for the finite sum back into the double sum, we obtain

(1-x)-n-1=m=0(m+nn)xm.

Q.E.D.

Title binomial formula for negative integer powers
Canonical name BinomialFormulaForNegativeIntegerPowers
Date of creation 2013-03-22 14:57:26
Last modified on 2013-03-22 14:57:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Corollary
Classification msc 26A06
Related topic GeneralizedBinomialCoefficients