Generating Function for Sequence of Powers of Constant

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Theorem

Let $c \in \R$ be a constant.

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

$\forall n \in \Z_{\ge 0}: a_n = c^n$

That is:

$\sequence {a_n} = 1, c, c^2, c^3, \ldots$


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

$\map G z = \dfrac 1 {1 - c z}$


Proof

Consider the sequence $\sequence {b_n}$ defined as:

$\forall n \in \Z_{\ge 0}: b_n = 1$

Let $\map H z$ be the generating function for $\sequence {b_n}$.


Then:

\(\ds \map H z\) \(=\) \(\ds \dfrac 1 {1 - z}\) Generating Function for Constant Sequence
\(\ds \leadsto \ \ \) \(\ds \map H {c z}\) \(=\) \(\ds \dfrac 1 {1 - c z}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \paren {b_n c^n} z^n\) Generating Function of Multiple of Parameter
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} c^n z^n\) as $b_n = 1$ throughout

The result follows from the definition of a generating function.

$\blacksquare$


Examples

Example: $\sequence {2^n}$

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

$\forall n \in \Z_{\ge 0}: a_n = 2^n$

That is:

$\sequence {a_n} = 1, 2, 4, 8, \ldots$


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

$\map G z = \dfrac 1 {1 - 2 z}$


Example: $\sequence {2^n + 3^n}$

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

$\forall n \in \Z_{\ge 0}: a_n = 2^n + 3^n$

That is:

$\sequence {a_n} = 2, 5, 13, 35, \ldots$


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

$\map G z = \dfrac 1 {1 - 2 z} + \dfrac 1 {1 - 3 z}$


Example: $\sequence {\paren {b + 1}^n - b^n}$

Let $b \in \R_{>0}$ be a positive real number.

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

$\forall n \in \Z_{\ge 0}: a_n = \paren {b + 1}^n - b^n$


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

$\map G z = \dfrac z {\paren {1 - b z} \paren {1 - b z - z} }$


Sources

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