In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like ${\displaystyle (1+x)^{n}}$ for a nonnegative integer ${\displaystyle n}$. Specifically, the binomial series is the Taylor series for the function ${\displaystyle f(x)=(1+x)^{\alpha }}$ centered at ${\displaystyle x=0}$, where ${\displaystyle \alpha \in \mathbb {C} }$ and ${\displaystyle |x|<1}$. Explicitly,

where the power series on the right-hand side of (1) is expressed in terms of the (generalized) binomial coefficients

${\displaystyle {\binom {\alpha }{k}}:={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.}$