generalized binomial coefficients
here may be any non-negative integer. Then Newton’s binomial series (http://planetmath.org/BinomialFormula) gets the form
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 when , and it presents such a branch (http://planetmath.org/GeneralPower) of the power (http://planetmath.org/GeneralPower) which gets the value 1 in the point .
In the case that is a non-negative integer and is great enough, one factor in the numerator of
vanishes, and hence the corresponding binomial coefficient 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.
and Vandermonde’s convolution
hold (the latter is proved by expanding the power to series). Cf. Pascal’s rule and Vandermonde identity.
|Title||generalized binomial coefficients|
|Date of creation||2013-03-22 14:41:53|
|Last modified on||2013-03-22 14:41:53|
|Last modified by||pahio (2872)|