binomial formula for negative integer powers

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

(1-x)^{-n}=\sum_{m=1}^{\infty}\binom{m+n-1}{n-1}x^{m}

Proof: We shall prove this by induction on $n$ . First, note that, if $n=1$ , then $\binom{m}{0}=1$ , so our formula reduces to

(1-x)^{-1}=\sum_{m=1}^{\infty}x^{m},

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}=\left(\sum_{k=0}^{\infty}x^{k}\right)\left(% \sum_{m=0}^{\infty}{m+n-1\choose n-1}x^{m}\right)

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

\sum_{m=0}^{\infty}\sum_{k=0}^{m}{n+k-1\choose n-1}x^{m}

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

\begin{matrix}x^{0}\sum_{m=0}^{\infty}\binom{m+n-1}{n-1}x^{m}=&\binom{n-1}{n-1% }&+&\binom{n}{n-1}x&+&\binom{n+1}{n-1}x^{2}&+&\binom{n+2}{n-1}x^{3}&+&\binom{n% +3}{n-1}x^{4}&+&\cdots\\ x^{1}\sum_{m=0}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&\binom{n-1}{n-1}x&+&\binom{% n}{n-1}x^{2}&+&\binom{n+1}{n-1}x^{3}&+&\binom{n+2}{n-1}x^{4}&+&\cdots\\ x^{2}\sum_{m=0}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&&&\binom{n-1}{n-1}x^{2}&+&% \binom{n}{n-1}x^{3}&+&\binom{n+1}{n-1}x^{4}&+&\cdots\\ x^{3}\sum_{m=0}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&&&&&\binom{n-1}{n-1}x^{3}&+% &\binom{n}{n-1}x^{4}&+&\cdots\\ .&.&.&.&.&.&.&.&.&.&.&.\end{matrix}

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.)

\sum_{k=0}^{m}\binom{n+k-1}{n-1}=\binom{m+n}{n}

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

(1-x)^{-n-1}=\sum_{m=0}^{\infty}\binom{m+n}{n}x^{m}.

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