Intuitive explanation for negative binomial expansion

Question

I understand the relationship between Pascal's triangle and binomial expansion for nonnegative integer powers by using combination e.g the binomial coefficient for $x^3$ in $(a + x)^8$ is equivalent to how many ways there are to choose 3 $x$'s from 8 possibilities.

I thought this idea would fail with regards to negative binomial expansion but I find out we can have cases where $r<0$ in $nCr$.

Can someone intuitively explain this? I kind of understand binomial expansion using Maclaurin series but I can't wrap my head around what negative combination is intuitively and why it works for negative binomial expansion.

Basically I don't understand the reason behind the combination formula still working for negative numbers.

Answer 1

The binomial identity \begin{align*} \color{blue}{\binom{n+1}{k+1}=\binom{n}{k+1}+\binom{n}{k}}\tag{1} \end{align*} is not only valid for natural numbers $n$ but also for $n\in\mathbb{Z}$ and $k\geq 0$ a natural number. We can use this relationship to extend the Pascal triangle to negative numbers $n$ shown in the table below. The numbers of the Pascal triangle for $n,k\geq 0$ follow by setting $\binom{n}{0}=1, \binom{0}{k}=\delta_{k,0}$. The entries with negative $n$ follow by extending the rule $\binom{n}{0}=1$ for $n\in\mathbb{Z}$. \begin{align*} \begin{array}{r|rrrrrrrrl} k&0&1&2&3&4&5&6&&\\ \hline n&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&&\\ -4&1&\color{blue}{-4}&\color{blue}{10}&-20&35&-56&84&&(1+x)^{-4}\\ -3&1&-3&\color{blue}{6}&-10&15&-21&28&&(1+x)^{-3}\\ -2&1&-2&3&-4&5&-6&7&&(1+x)^{-2}\\ -1&1&-1&1&-1&1&-1&1&&(1+x)^{-1}\\ 0&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^0\\ 1&1&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^1\\ 2&1&2&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^2\\ 3&1&\color{blue}{3}&\color{blue}{3}&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^3\\ 4&1&4&\color{blue}{6}&4&1&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^4\\ \end{array} \end{align*}

When looking for instance at the last line of the table above we see the number $6$ is according to (1) the sum of the entry $3$ above $6$ with its left neighbor $3$. \begin{align*} 6=3+3\qquad\qquad \binom{4}{2}=\binom{3}{1}+\binom{3}{2} \end{align*} The algebraic connection is given by \begin{align*} \color{blue}{(1+x)^4=(1+x)(1+x)^3} \end{align*} or more detailed with $[x^k]$ denoting the coefficient of $x^k$: \begin{align*} \binom{4}{2}&=[x^2](1+x)^4\\ &=[x^2](1+x)(1+x)^3\\ &=\left([x^2]+[x^1]\right)(1+x)^3\\ &=\binom{3}{2}+\binom{3}{1}\tag{2} \end{align*} The algebraic connection holds also for negative $n$, for instance: \begin{align*} \color{blue}{(1+x)^{-3}=(1+x)(1+x)^{-4}} \end{align*} and we obtain similarly to (2): \begin{align*} 6=10+\left(-4\right)\qquad\qquad \binom{-3}{2}=\binom{-4}{2}+\binom{-4}{1} \end{align*} or more detailed with $[x^k]$ denoting the coefficient of $x^k$: \begin{align*} \binom{-3}{2}&=[x^2](1+x)^{-3}\\ &=[x^2](1+x)(1+x)^{-4}\\ &=\left([x^2]+[x^1]\right)(1+x)^{-4}\\ &=\binom{-4}{2}+\binom{-4}{1}\tag{2} \end{align*}

We can also see in the table the binomial identity \begin{align*} \binom{-n}{k}=\binom{n+k-1}{k}(-1)^k \end{align*} is given as rotation of a somewhat modified Pascal triangle. \begin{align*} \begin{array}{r|rrrrrrrrl} k&0&1&2&3&4&5&6&&\\ \hline n&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&\color{white}{-0}&&\\ -4&\color{blue}{1}&-4&10&-20&35&-56&84&&(1+x)^{-4}\\ -3&\color{blue}{1}&\color{blue}{-3}&6&-10&15&-21&28&&(1+x)^{-3}\\ -2&\color{blue}{1}&\color{blue}{-2}&\color{blue}{3}&-4&5&-6&7&&(1+x)^{-2}\\ -1&\color{blue}{1}&\color{blue}{-1}&\color{blue}{1}&\color{blue}{-1}&1&-1&1&&(1+x)^{-1}\\ 0&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^0\\ 1&1&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^1\\ 2&1&2&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^2\\ 3&1&3&3&1&\color{lightgrey}{0}&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^3\\ 4&1&4&6&4&1&\color{lightgrey}{0}&\color{lightgrey}{0}&&(1+x)^4\\ \end{array} \end{align*}

Answer 2

$(1+x)^{-n}$ can also be written as $\frac{1}{(1+x)^n}$.

The denominator can be expanded and then viewed as the usual identity $$\frac1{1-x}=1+x+x^2+\dots$$

For example with $n=2$:

$$1+(-2x-x^2)+(2x+x^2)^2+(-2x-x^2)^3+\dots$$

This can have the like terms collected, which gives:

$$\binom{-2}{1}=-2$$ $$\binom{-2}{2}=3$$

etc...

Answer 3

The Question

In this answer, it is shown that $$ \binom{-n}{j}=(-1)^j\binom{n+j-1}{j}\tag1 $$ If I understand correctly, the question is asking why, when $(1)$ is applied to the Binomial Theorem, we get $(1+x)^{-n}$.

Prior Results

In this answer are three proofs of this cancellation formula $$ \sum_{j=k}^n(-1)^{j-k}\binom{n}{j}\binom{j}{k} =[n=k]\tag2 $$ where $[\cdots]$ are Iverson Brackets.

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Verification of $\bf{(1)}$ Using the Cauchy Product Formula

For $k\ge1$, we have $$ \begin{align} \sum_{j=0}^k\binom{-n}{j}\binom{n}{k-j} &=\sum_{j=0}^k(-1)^j\binom{n+j-1}{n-1}\binom{n}{n+j-k}\tag{3a}\\ &=\sum_{j=0}^k(-1)^j\sum_{i=0}^{k-1}\binom{n+j-k}{i}\binom{k-1}{n-1-i}\binom{n}{n+j-k}\tag{3b}\\ &=\sum_{i=0}^{k-1}(-1)^{n-k-i}\binom{k-1}{n-1-i}[n=i]\tag{3c}\\[9pt] &=0\tag{3d} \end{align} $$ Explanation:
$\text{(3a)}$: apply $(1)$ and the symmetry of Pascal's Triangle
$\text{(3b)}$: Vandermonde's Identity
$\text{(3c)}$: apply $(2)$
$\text{(3d)}$: $\binom{k-1}{-1}=0$

If $k=0$, the sum is $\binom{-n}{0}\binom{n}{0}=1$. Therefore, using $(1)$, $(2)$, and Vandermonde, we have shown $$ \sum_{j=0}^k\binom{-n}{j}\binom{n}{k-j}=[k=0]\tag4 $$ which by the Cauchy Product Formula verifies that $$ \underbrace{\sum_{j=0}^\infty\binom{-n}{j}x^j}_{(1+x)^{-n}}\ \underbrace{\sum_{j=0}^n\binom{n}{j}x^j}_{(1+x)^n}=1\tag5 $$

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Verification of $\bf{(1)}$ Using Induction

Using the formula for the sum of a geometric series, we have $$ (1+x)^{-1}=\sum_{k=0}^\infty(-1)^kx^k\tag6 $$ We can verify $(1)$ inductively using $(6)$ and the Cauchy Product Formula. Assume that $(1)$ is true for a given $n$, then $$ \begin{align} (1+x)^{-n-1} &=\color{#C00}{(1+x)^{-1}}\color{#090}{(1+x)^{-n}}\tag{7a}\\[9pt] &=\sum_{k=0}^\infty\sum_{j=0}^k\color{#C00}{(-1)^{k-j}x^{k-j}}\color{#090}{(-1)^j\binom{n+j-1}{j}x^j}\tag{7b}\\ &=\sum_{k=0}^\infty\sum_{j=0}^k(-1)^k\binom{n+j-1}{j}x^k\tag{7c}\\ &=\sum_{k=0}^\infty(-1)^k\binom{n+k}{k}x^k\tag{7d}\\ \end{align} $$ Explanation:
$\text{(7b)}$: $(6)$ and the inductive hypothesis
$\text{(7c)}$: collect terms
$\text{(7d)}$: Hockey-Stick Identity

which verifies $(1)$ for $n+1$.