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.
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