Editing Binomial coefficient (section)

=== Generalization and connection to the binomial series ===
{{Main|Binomial series}}
The multiplicative formula allows the definition of binomial coefficients to be extended<ref>See {{Harv|Graham|Knuth|Patashnik|1994}}, which also defines <math>\tbinom n k = 0</math> for <math>k<0</math>. Alternative generalizations, such as to [[#Two real or complex valued arguments|two real or complex valued arguments]] using the [[Gamma function]] assign nonzero values to <math>\tbinom n k</math> for <math>k < 0</math>, but this causes most binomial coefficient identities to fail, and thus is not widely used by the majority of definitions. One such choice of nonzero values leads to the aesthetically pleasing "Pascal windmill" in Hilton, Holton and Pedersen, ''Mathematical reflections: in a room with many mirrors'', Springer, 1997, but causes even [[Pascal's identity]] to fail (at the origin).</ref> by replacing ''n'' by an arbitrary number ''α'' (negative, real, complex) or even an element of any [[commutative ring]] in which all positive integers are invertible:
<math display="block">\binom \alpha k = \frac{\alpha^{\underline k}}{k!} = \frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha-k+1)}{k(k-1)(k-2)\cdots 1}
\quad\text{for } k\in\N \text{ and arbitrary } \alpha.
</math>

With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the <math>\tbinom\alpha k</math> binomial coefficients:
{{NumBlk2|:|<math> (1+X)^\alpha = \sum_{k=0}^\infty {\alpha \choose k} X^k.</math>|2}}

This formula is valid for all complex numbers ''α'' and ''X'' with |''X''|&nbsp;&lt;&nbsp;1. It can also be interpreted as an identity of [[formal power series]] in ''X'', where it actually can serve as definition of arbitrary powers of [[power series]] with constant coefficient equal to&nbsp;1; the point is that with this definition all identities hold that one expects for [[exponentiation]], notably
<math display="block">(1+X)^\alpha(1+X)^\beta=(1+X)^{\alpha+\beta} \quad\text{and}\quad ((1+X)^\alpha)^\beta=(1+X)^{\alpha\beta}.</math>

If ''α'' is a nonnegative integer ''n'', then all terms with {{math|''k'' &gt; ''n''}} are zero,<ref>When <math>\alpha = n</math> is a nonnegative integer, <math>\textstyle \binom{n}{k} = 0</math> for <math>k > n</math> because the <math>(k = n+1)</math>-th factor of the numerator is <math>n - (n+1) + 1 = 0</math>. Thus, the <math>k</math>-th term is a [[Zero-product property|zero product]] for all <math>k \geq n + 1</math>.</ref> and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of ''α'', including negative integers and rational numbers, the series is really infinite.