nLab binomial theorem

Context

Arithmetic

Combinatorics

Statement
Combinatorial interpretation
Pascal’s triangle
Related concepts
References

Statement

For $k$ a natural number and $r$ a complex number, define the falling factorial by

r^{\underline{k}} = r(r-1)\ldots (r-k+1),

a polynomial of degree $k$ evaluated at $r$ . If $r$ is a natural number, this expression vanishes for $k \gt r$ .

The binomial theorem may be stated thus: if $r$ is any complex number and ${|x|} \lt 1$ , then

(1 + x)^r = \sum_{k \geq 0} \frac{r^{\underline{k}} x^k}{k!}

where the left side may be formally defined as $\exp(r \cdot \log (1+x))$ , taking the principal branch of the logarithm as defined by the power series

\log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots

with radius of convergence equal to $1$ . (A formal verification of the binomial theorem may be found at coinduction.)

Thus, if we define the binomial coefficient $\binom{r}{k}$ by the formula

\binom{r}{k} \coloneqq \frac{r^{\underline{k}}}{k!},

then we have

(y + x)^r \;=\; \sum_{k \geq 0} \binom{r}{k} \cdot y^{r-k} \cdot x^k

whenever ${|\frac{x}{y}|} \lt 1$ , i.e., whenever ${|y|} \gt {|x|}$ . More precisely: for any fixed $y \neq 0$ , this equation holds for any branch of the logarithm that we use to define $(y+x)^r$ as $\exp(r\log(y+x))$ over the domain $\{x: {|x|} \lt {|y|}\}$ .

Combinatorial interpretation

The special finitary case in which $i, j, n$ are positive integers,

(i + j)^n = \sum_{0 \leq k \leq n} \binom{n}{k} i^k j^{n-k}

may be established combinatorially (or in “bijective fashion”) as follows. Start by interpreting $(i + j)^n$ as the number of functions $f: N \to I \sqcup J$ from an $n$ -element set, where $I, J$ have cardinalities $i, j$ respectively. By pulling back $f$ along each of the inclusions of $I, J$ into $I \sqcup J$ , we get functions $f_I: f^{-1}(I) \to I$ , $f_J: f^{-1}(J) \to J$ . Here $f^{-1}(I)$ and $f^{-1}(J)$ are complementary subsets of $N$ , say of cardinalities $k$ and $n-k$ respectively. (In effect, we are using the fact that the category of finite sets is an extensive category.) Thus, $f$ determines and is uniquely determined by the following data

A subset $K (= f^{-1}(I))$ of $N$ ,
A function $g (= f_I)$ of the form $K \to I$ ,
A function $h (= f_J)$ of the form $N-K \to J$

and by counting the number of such triplets $(K, g, h)$ , we are led to the right-hand side of the previous displayed equation.

Notice this gives a rigorous proof for the polynomial identity

(x+y)^n = \sum_{0 \leq k \leq n} \binom{n}{k} x^k y^{n-k}

since a polynomial in $\mathbb{Z}[x, y]$ is the zero polynomial if it vanishes for all positive integer values substituted for $x$ and $y$ .

Pascal’s triangle

The binomial coefficient polynomials $\binom{x}{k}$ (here $x$ is an indeterminate) satisfy the recurrence

\Delta \binom{x}{k} \coloneqq \binom{x+1}{k} - \binom{x}{k} = \binom{x}{k-1}, \qquad \binom{x}{0} \coloneqq 1, \binom{0}{k} = 0\; (k \neq 0)

where the first two equations may also be written as

\Delta \frac{x^\underline{k}}{k!} = \frac{x^\underline{k-1}}{(k-1)!}, \qquad \frac{x^\underline{0}}{0!} = 1.

The first equation may be viewed as the discrete analogue of the continuous derivative formula

\frac{d}{d x} \frac{x^k}{k!} = \frac{x^{k-1}}{(k-1)!}.

The more familiar form of this recurrence,

\binom{x+1}{k} = \binom{x}{k} + \binom{x}{k-1},

may be interpreted combinatorially: a $k$ -element subset $K$ of $X + 1$ is either entirely contained in $X$ , or is determined by the $(k-1)$ -element subset of $X$ that is $K \cap X$ .