Difference between revisions of "Newton binomial"
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''binomium of Newton'' | ''binomium of Newton'' | ||
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the terms of the binomial: | The formula for the expansion of an arbitrary positive integral power of a [[Binomial|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 | where | ||
− | + | $$ | |
+ | \left ( | ||
+ | \begin{array}{c} | ||
+ | m \\ | ||
+ | k | ||
+ | \end{array} | ||
+ | |||
+ | \right ) = \ | ||
− | + | \frac{m! }{k! ( m - k)! } | |
− | + | $$ | |
− | + | are the [[Binomial coefficients|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|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==== | ====Comments==== | ||
The coefficients | 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. | are called multinomial coefficients. |
Latest revision as of 08:02, 6 June 2020
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.
Newton binomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton_binomial&oldid=47965