binomial formula for negative integer powers

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


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


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


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


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


To evaluate the finite sums, we shall use the following identity for binomial coefficients. (See the entry“binomial coefficient” for more information about this identity.)


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



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