Editing Multiset (section)

=== Generalization and connection to the negative binomial series ===
The multiplicative formula allows the definition of multiset coefficients to be extended by replacing {{mvar|n}} by an arbitrary number {{mvar|α}} (negative, [[real number|real]], or complex):
<math display="block">\left(\!\!{\alpha\choose k}\!\!\right) = \frac{\alpha^{\overline 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 negative binomial formula (with one of the variables set to 1), which justifies calling the <math>\left(\!\!{\alpha\choose k}\!\!\right)</math> negative binomial coefficients:
<math display="block">(1-X)^{-\alpha} = \sum_{k=0}^\infty \left(\!\!{\alpha\choose k}\!\!\right) X^k.</math>

This [[Taylor series]] formula is valid for all complex numbers ''α'' and ''X'' with {{math|{{abs|''X''}} < 1}}. It can also be interpreted as an [[identity (mathematics)|identity]] of [[formal power series]] in ''X'', where it actually can serve as definition of arbitrary powers of 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>
and formulas such as these can be used to prove identities for the multiset coefficients.

If {{mvar|α}} is a nonpositive integer {{mvar|n}}, then all terms with {{math|''k'' > −''n''}} are zero, and the infinite series becomes a finite sum. However, for other values of {{mvar|α}}, including positive integers and [[rational number]]s, the series is infinite.