PlanetMath: generalized binomial coefficients

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generalized binomial coefficients

(Definition)

The binomial coefficients

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

(1)

where

is a non-negative integer and $r \in \{0,\,1,\,2,\,\ldots,\,n\}$ , can be generalized for all integer and non-integer values of

by using the reduced form

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

(2)

here

may be any non-negative integer. Then Newton's binomial series gets the simple form

$\displaystyle (1\!+\!z)^{\alpha} = \sum_{r = 0}^{\infty}{\alpha\choose r}z^r = 1\!+\!{\alpha\choose1}z\!+\!{\alpha\choose 2}z^2\!+\cdots$

(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 $\vert z\vert < 1$ , and it presents such a branch of the power $(1\!+\!z)^{\alpha}$ which gets the value 1 in the point

In the case that $\alpha$ is a non-negative integer and 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

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 , , 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.

"generalized binomial coefficients" is owned by pahio.

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Also defines:

Pascal's formula, Vandermonde's formula

This object's parent.

Attachments:: Taylor series of arcus sine (Example) by pahio

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Cross-references: Vandermonde identity, Pascal's rule, convolution, complex, binomial theorem, sum, finite, vanishes, numerator, factor, point, complex number, series, radius of convergence, integer, binomial coefficients
There are 2 references to this entry.

This is version 23 of generalized binomial coefficients, born on 2004-10-06, modified 2006-10-06.
Object id is 6309, canonical name is GeneralizedBinomialCoefficients.
Accessed 8973 times total.

Classification:

AMS MSC:	05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions)
	11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)

Pending Errata and Addenda

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