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generalized binomial coefficients
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(Definition)
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The binomial coefficients
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(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 form
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here $r$ may be any non-negative integer. Then Newton's binomial series gets the simple form
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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 of the power $(1\!+\!z)^{\alpha}$ 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
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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
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and Vandermonde's convolution
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hold (the latter is proved by expanding the power $(1\!+\!z)^{\alpha+\beta}$ to series). Cf. Pascal's rule and Vandermonde identity.
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"generalized binomial coefficients" is owned 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 11910 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) |
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Pending Errata and Addenda
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