Generating Function for Binomial Coefficients

Theorem

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N: a_n = \begin{cases}

\dbinom m n & : n = 0, 1, 2, \ldots, m \\ 0 & : \text{otherwise}\end{cases}$

where $\dbinom m n$ denotes a binomial coefficient.

Then the generating function for $\sequence {a_n}$ is given as:

$\ds \map G z = \sum_{n \mathop = 0}^m \dbinom m n z^n = \paren {1 + z}^m$

Proof

\(\ds \paren {1 + z}^m\)

\(=\)

\(\ds \sum_{n \mathop = 0}^m \binom m n z^n\)

Binomial Theorem

The result follows from the definition of a generating function.

$\blacksquare$

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.1$: Generating functions: Example $3$