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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
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