Newton binomial

binomium of Newton

The formula for the expansion of an arbitrary positive integral power of a binomial in a polynomial arranged in powers of one of the terms of the binomial:

$$ \tag{* } ( z _ {1} + z _ {2} ) ^ {m\ } = $$

$$ = \ z _ {1} ^ {m} + { \frac{m}{1!} } z _ {1} ^ {m - 1 } z _ {2} + \frac{m ( m - 1) }{2! } z _ {1} ^ {m - 2 } z _ {2} ^ {2} + \dots + z _ {2} ^ {m\ } = $$

$$ = \ \sum _ {k = 0 } ^ { m } \left ( \begin{array}{c} m \\ k \end{array} \right ) z _ {1} ^ {m - k } z _ {2} ^ {k} , $$

where

$$ \left ( \begin{array}{c} m \\ k \end{array} \right ) = \ \frac{m! }{k! ( m - k)! } $$

are the binomial coefficients. For $ n $ terms formula (*) takes the form

$$ ( z _ {1} + \dots + z _ {n} ) ^ {m\ } = $$

$$ = \ \sum _ {k _ {1} + \dots + k _ {n} = m } \frac{m! }{k _ {1} ! \dots k _ {n} ! } z _ {1} ^ {k _ {1} } \dots z _ {n} ^ {k _ {n} } . $$

For an arbitrary exponent $ m $, real or even complex, the right-hand side of (*) is, generally speaking, a binomial series.

The gradual mastering of binomial formulas, beginning with the simplest special cases (formulas for the "square" and the "cube of a sum" ) can be traced back to the 11th century. I. Newton's contribution, strictly speaking, lies in the discovery of the binomial series.

Comments

The coefficients

$$ \left ( \begin{array}{c} m \\ k _ {1} \dots k _ {n} \end{array} \right ) = \ \frac{m! }{k _ {1} ! \dots k _ {n} ! } ,\ \ k _ {1} + \dots + k _ {n} = m, $$

are called multinomial coefficients.