generalized binomial coefficients

The binomial coefficients

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)={\displaystyle \frac{n!}{(n-r)!r!}},$ (1)

where $n$ is a non-negative integer and  $r\in \{0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots },n\}$,  can be generalized for all integer and non-integer values of $n$ by using the reduced (http://planetmath.org/Division) form

$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)={\displaystyle \frac{n(n-1)(n-2)\mathrm{\dots }(n-r+1)}{r!}};$ (2)

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

${(1+z)}^{\alpha }={\displaystyle \sum _{r=0}^{\mathrm{\infty }}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{r}}\right)z^r=1+\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{1}}\right)z+\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{2}}\right)z^2+\mathrm{\cdots }$ (3)

It is not hard to show that the radius of convergence of this series is 1.  This series expansion is valid for every complex number $\alpha$ when  ,  and it presents such a branch (http://planetmath.org/GeneralPower) of the power (http://planetmath.org/GeneralPower) ${(1+z)}^{\alpha }$ which gets the value 1 in the point  $z=0$.

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

$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{r}}\right)={\displaystyle \frac{\alpha (\alpha -1)(\alpha -2)\mathrm{\dots }(\alpha -r+1)}{r!}}$ (4)

vanishes, and hence the corresponding binomial coefficient $\left(\genfrac{}{}{0pt}{}{\alpha }{r}\right)$ 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 theorem.

For all complex values of $\alpha$, $\beta$ and non-negative integer values of $r$, $s$, the Pascal’s formula

$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{r}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{r+1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +1}{r+1}}\right)$ (5)

and Vandermonde’s convolution

${\displaystyle \sum _{r=0}^s}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{r}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{s-r}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +\beta }{s}}\right)$ (6)

hold (the latter is proved by expanding the power ${(1+z)}^{\alpha +\beta }$ to series).  Cf. Pascal’s rule and Vandermonde identity.

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