## A note on Fibonacci quaternions.(English)Zbl 0191.32701

We define the $$n$$-th Fibonacci quaternion $$Q_n$$ and the $$n$$-th generalized Fibonacci quaternion $$P_n$$ as follows:
$Q_n = F_n + iF_{n+1} + jF_{n+2} + kF_{n+3}$
where $$F_n$$ is the $$n$$-th Fibonacci number,
$P_n = H_n + iH_{n+1} + jH_{n+2} + kH_{n+3}$
where $$H_n$$ is the $$n$$-th generalized Fibonacci number. We know that $$H_n = pF_n + qF_{n-1}$$ where $$p$$ and $$q$$ are numbers satisfying $$H_1 = p$$, $$H_2 = p + q$$. In this note the author derives some relations connecting these two quaternions. Some of the results obtained are as follows:
$P_n=pQ_n+qQ_{n-1}, \tag{i}$
$P_n\overline Q_n - \overline P_nQ_n = 2 [F_n P_n - H_nQ_n], \tag{ii}$
$P_n\overline Q_n + \overline P_nQ_n =2 [P_nF_n + Q_nH_n - P_nQ_n], \tag{iii}$
$P_nQ_n -\overline P_n\overline Q_n = 2 [P_nF_n + Q_nH_n - 2H_nF_n, \tag{iv}$
also the theorem $Q_{n-1}^2 + Q_n^2=2Q_{2n-1} -3 L_{2n+2}$ is proved. In the above results $$\overline P_n$$ and $$\overline Q_n$$, are conjugate quaternions respectively of $$P_n$$ and $$Q_n$$.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations

### Keywords:

Fibonacci quaternions
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