Binomial theorem

For other topics using the name "binomial", see binomial (disambiguation).

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}\quad \quad \quad (1)$

whenever n is any non-negative integer, the numbers

${n \choose k}={\frac {n!}{k!(n-k)!}}$

are the binomial coefficients, and $n!$ denotes the factorial of n.

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to Chinese mathematician Yang Hui in the 13th century.

For example, here are the cases n = 2, n = 3 and n = 4:

$(x+y)^{2}=x^{2}+2xy+y^{2}\,$

$(x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}\,$

$(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.\,$

Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a semiring as long as xy = yx.

Newton's generalized binomial theorem

Isaac Newton generalized the formula to other exponents by considering an infinite series:

${(x+y)^{r}=\sum _{k=0}^{\infty }{r \choose k}x^{k}y^{r-k}\quad \quad \quad (2)}$

where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by

${r \choose k}={1 \over k!}\prod _{n=0}^{k-1}(r-n)={\frac {r(r-1)(r-2)\cdots (r-(k-1))}{k!}}\,$

In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not appear.

Another way to express this quantity is

${r \choose k}={\frac {(-1)^{k}}{k!}}(-r)_{k},$

which is important when one is working with infinite series and would like to represent them in terms of Generalized Hypergeometric Functions. The notation $(\cdot )_{k}$ represents the Pochhammer symbol. This form is vital in applied mathematics, for example, when evaluating the formulas that model the statistical properties of the phase-front curvature of a light wave as it propagates through optical atmospheric turbulence.

A particularly handy but non-obvious form holds for the reciprocal power:

${\frac {1}{(1-x)^{r}}}=\sum _{k=0}^{\infty }{r+k-1 \choose r-1}x^{k}.$

For a more extensive account of Newton's generalized binomial theorem, see binomial series.

The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute value | x/y | is less than one.

The geometric series is a special case of (2) where we choose y = 1 and r = −1.

Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, y is invertible and ||x/y|| < 1.

"Binomial type"

The binomial theorem can be stated by saying that the polynomial sequence

$\left\{\,x^{k}:k=0,1,2,\dots \,\right\}\,$

is of binomial type.

Proof (inductive)

When $n=1$,

$(a+b)^{1}=\sum _{k=0}^{1}{1 \choose k}a^{1-k}b^{k}={1 \choose 0}a^{1}b^{0}+{1 \choose 1}a^{0}b^{1}=a+b$.

For the inductive step, assume it holds for $m$. Then for n = $m+1$,

$(a+b)^{m+1}$ $=$ $a(a+b)^{m}+b(a+b)^{m}$

$=$ $a\sum _{k=0}^{m}{m \choose k}a^{m-k}b^{k}+b\sum _{j=0}^{m}{m \choose j}a^{m-j}b^{j}$ by the inductive hypothesis

$=$ $\sum _{k=0}^{m}{m \choose k}a^{m-k+1}b^{k}+\sum _{j=0}^{m}{m \choose j}a^{m-j}b^{j+1}$ by multiplying through by $a$ and $b$

$=$ $a^{m+1}+\sum _{k=1}^{m}{m \choose k}a^{m-k+1}b^{k}+\sum _{j=0}^{m}{m \choose j}a^{m-j}b^{j+1}$ by pulling out the $k=0$ term

$=$ $a^{m+1}+\sum _{k=1}^{m}{m \choose k}a^{m-k+1}b^{k}+\sum _{k=1}^{m+1}{m \choose k-1}a^{m-k+1}b^{k}$ by letting $j=k-1$

$=$ $a^{m+1}+\sum _{k=1}^{m}{m \choose k}a^{m-k+1}b^{k}+\sum _{k=1}^{m}{m \choose k-1}a^{m+1-k}b^{k}+b^{m+1}$ by pulling out the $k=m+1$ term from the RHS

$=$ $a^{m+1}+b^{m+1}+\sum _{k=1}^{m}\left[{m \choose k}+{m \choose k-1}\right]a^{m+1-k}b^{k}$ by combining the sums

$=$ $a^{m+1}+b^{m+1}+\sum _{k=1}^{m}{m+1 \choose k}a^{m+1-k}b^{k}$ from Pascal's rule

$=$ $\sum _{k=0}^{m+1}{m+1 \choose k}a^{m+1-k}b^{k}$ by adding in the $m+1$ terms,

as desired.

See also

• multinomial theorem
• Pascal's triangle

inductive proof of binomial theorem at PlanetMath.