Generating Function for Natural Numbers/Corollary
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Theorem
Let $\sequence {a_n}$ be the sequence defined as:
- $\forall n \in \N_{> 0}: a_n = n - 1$
That is:
- $\sequence {a_n} = 1, 2, 3, 4, \ldots$
Then the generating function for $\sequence {a_n}$ is given as:
- $H \paren z = \dfrac z {\paren {1 - z}^2}$
Proof
From Generating Function for Natural Numbers:
- $\sequence {a_n} = 0, 1, 2, 3, 4, \ldots$
has the generating function:
- $G \paren z = \dfrac 1 {\paren {1 - z}^2}$
Then by Generating Function by Power of Parameter:
- $z G \paren z = \dfrac z {\paren {1 - z}^2}$
is the generating function for the sequence defined as:
- $\forall n \in \N_{> 0}: a_{n - 1} = n - 1$
that is:
- $0, a_1, a_2, \ldots$
where:
- $a_1 = 1, a_2 = 2, \ldots$
That is:
- $\sequence {a_n} = 1, 2, 3, 4, \ldots$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-4}$ Generating Functions: Exercise $1$