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 ( 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 ( 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 ( of the power ( (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.

Title generalized binomial coefficients
Canonical name GeneralizedBinomialCoefficients
Date of creation 2013-03-22 14:41:53
Last modified on 2013-03-22 14:41:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Definition
Classification msc 11B65
Classification msc 05A10
Related topic BinomialFormula
Related topic GeneralPower
Related topic BinomialFormulaForNegativeIntegerPowers
Defines Pascal’s formula
Defines Vandermonde’s formula