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Part BCSIR NET June 2023generating-function

Generating function

Consider the function f defined by $f (z) = 1/ (1 - z - z^{2})$ for $z \in C$ such that $1 - z - z^{2} \neq = 0$ . Which of the following statements is true?

A.f is an entire function.
B.f has a simple pole at z = 0.
C.f has a Taylor series expansion $f (z) = \sum a_{n} z^{n}$ , where $a_{0} = 1, a_{1} = 0$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.
D.f has a Taylor series expansion $f (z) = \sum a_{n} z^{n}$ , where $a_{0} = 1, a_{1} = 1$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.✓

Solution

$(1 - z - z^{2}) \sum a_{n} z^{n} = 1$ gives $a_{0} = 1, a_{1} - a_{0} = 0 \Rightarrow a_{1} = 1$ , and $a_{n}$ ₊ $_{2} = a_{n}$ ₊ $_{1} + a_{n}$ : the Fibonacci generating function. Poles are at the roots of $1 - z - z^{2}$ , not at 0.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Zeros of a non-constant holomorphic function cannot accumulate

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