First, I want to define the identities as





And then I want to use these identities in my sequence

$Q_n = F_n + iF_{n+1} + j F_{n+2} + k F_{n+3}$,

where $F_n$ is the Fibonacci sequence and $n \geq 0$.

For example I want to calculate and simplify all results for $Q_{11} Q_{9} - Q_{10}^2 = ?$.

With my best.

  • $\begingroup$ Look at the Quaternions package. $\endgroup$
    – Szabolcs
    Feb 19, 2018 at 21:01
  • 3
    $\begingroup$ Could just use the matrix representation from here or here $\endgroup$ Feb 19, 2018 at 22:18

1 Answer 1


You can use the Quaternions package. Load:


Define your sequence:

Q[n_] := Fibonacci[n] + I Fibonacci[n+1] + J Fibonacci[n+2] + K Fibonacci[n+3]

Then your example:

Q[11]**Q[9] - Q[10]**Q[10]

2 + 2 J + 5 K


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