lacunary - Mathnotes

Definition: Binomial Coefficient \@{binomial-coefficient}

For n,k∈N,n≥k≥0, the binomial coefficient is the coefficient of the xk term in the polynomial expansion of the binomial power (1+x)n. It is denoted as (nk) and can be expressed through @factorial notation as

(nk)=n!k!(n−k)!.

Definition: Generalized Binomial Coefficient \@{generalized-binomial-coefficient}

For any complex number n∈C and any integer k≥0, the generalized binomial coefficient is defined by

(nk)=n(n−1)(n−2)⋯(n−k+1)k!.

Theorem: Binomial Theorem \@{binomial-theorem}

The expansion of any nonnegative integer power n of the @binomial x+y is a sum of the form

(x+y)n=∑k=0n(nk)xn−kyk=∑k=0n(nk)xkyn−k.

Referenced by (2 direct)

Direct references:

Remark \@{remark-4}

A special case of the Binomial Theorem is the case where y=1:

(x+1)n=∑k=0n(nk)xk.