# generalized binomial coefficients

 $\displaystyle{n\choose r}=\frac{n!}{(n\!-\!r)!r!},$ (1)

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

 $\displaystyle{n\choose r}=\frac{n(n\!-\!1)(n\!-\!2)\ldots(n\!-\!r\!+\!1)}{r!};$ (2)

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

 $\displaystyle(1\!+\!z)^{\alpha}=\sum_{r=0}^{\infty}{\alpha\choose r}z^{r}=1\!+% \!{\alpha\choose 1}z\!+\!{\alpha\choose 2}z^{2}\!+\cdots$ (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  $|z|<1$,  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

 $\displaystyle{\alpha\choose r}=\frac{\alpha(\alpha\!-\!1)(\alpha\!-\!2)\ldots(% \alpha\!-\!r\!+\!1)}{r!}$ (4)

vanishes, and hence the corresponding binomial coefficient ${\alpha\choose r}$ 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

 $\displaystyle{\alpha\choose r}\!+\!{\alpha\choose r\!+\!1}={{\alpha\!+\!1}% \choose{r\!+\!1}}$ (5)

and Vandermonde’s convolution

 $\displaystyle\sum_{r=0}^{s}{\alpha\choose r}\!{\beta\choose{s\!-\!r}}={{\alpha% \!+\!\beta}\choose s}$ (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