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单词 GeneralizedBinomialCoefficients
释义

generalized binomial coefficients


The binomial coefficientsMathworldPlanetmath

(nr)=n!(n-r)!r!,(1)

where n is a non-negative integer and  r{0, 1, 2,,n}, can be generalized for all integer and non-integer values of n by using the reduced (http://planetmath.org/Division) form

(nr)=n(n-1)(n-2)(n-r+1)r!;(2)

here r may be any non-negative integer.  Then Newton’s binomial series (http://planetmath.org/BinomialFormula) gets the form

(1+z)α=r=0(αr)zr=1+(α1)z+(α2)z2+(3)

It is not hard to show that the radius of convergenceMathworldPlanetmath of this series is 1.  This series expansion is valid for every complex numberPlanetmathPlanetmath α when  |z|<1,  and it presents such a branch (http://planetmath.org/GeneralPower) of the power (http://planetmath.org/GeneralPower) (1+z)α which gets the value 1 in the point  z=0.

In the case that α is a non-negative integer and r is great enough, one factor in the numerator of

(αr)=α(α-1)(α-2)(α-r+1)r!(4)

vanishes, and hence the corresponding binomial coefficient (αr) equals to zero; accordingly also all following binomial coefficients with a greater r are equal to zero.  It means that the series is left to being a finite sum, which gives the binomial theoremMathworldPlanetmath.

For all complex values of α, β and non-negative integer values of r, s, the Pascal’s formulaMathworldPlanetmathPlanetmath

(αr)+(αr+1)=(α+1r+1)(5)

and Vandermonde’s convolution

r=0s(αr)(βs-r)=(α+βs)(6)

hold (the latter is proved by expanding the power (1+z)α+β to series).  Cf. Pascal’s rule and Vandermonde identityMathworldPlanetmath.

Titlegeneralized binomial coefficients
Canonical nameGeneralizedBinomialCoefficients
Date of creation2013-03-22 14:41:53
Last modified on2013-03-22 14:41:53
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id26
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 11B65
Classificationmsc 05A10
Related topicBinomialFormula
Related topicGeneralPower
Related topicBinomialFormulaForNegativeIntegerPowers
DefinesPascal’s formula
DefinesVandermonde’s formula
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