On higher order Horadam 3-parameter generalized quaternions
Gamaliel Morales
TL;DR
The paper extends higher order Horadam numbers to a 3-parameter generalized quaternion setting by defining \\mathbf{Q}W_n(s)=W_n(s)\\mathbf{1}+W_{n+1}(s)\\mathbf{i}+W_{n+2}(s)\\mathbf{j}+W_{n+3}(s)\\mathbf{k} and \\mathbf{Q}U_n(s)=U_n(s)\\mathbf{1}+U_{n+1}(s)\\mathbf{i}+U_{n+2}(s)\\mathbf{j}+U_{n+3}(s)\\mathbf{k}, establishing the induced recurrence \\mathbf{Q}W_{n+2}(s)=(\\alpha^s+\\beta^s)\\mathbf{Q}W_{n+1}(s)-(\\alpha\\beta)^s\\mathbf{Q}W_n(s) and deriving Binet-type, generating function, and summation identities. The authors derive explicit quaternionic Binet formulas \\mathbf{Q}W_n(s)=\frac{A\\alpha^{sn}\\Theta_\alpha(s)-B\\beta^{sn}\\Theta_\beta(s)}{A\\alpha^s-B\\beta^s} with \\Theta_\alpha(s)=\\mathbf{1}+\\alpha^s\\mathbf{i}+\\alpha^{2s}\\mathbf{j}+\\alpha^{3s}\\mathbf{k} and analogous for \\Theta_\beta(s), as well as Catalan, Cassini, and d'Ocagne-type identities expressed through these theta-quaternions. Additional results include summation formulas, cross-type product identities with \\mathbf{Q}U_n(s), generating functions, exponential generating functions, and a matrix generator that encodes the sequences, collectively providing a comprehensive algebraic framework for these generalized quaternion sequences.
Abstract
Recently, Kuloğlu {\it et al.} \cite{Kul} introduced the higher order Horadam numbers. In this study, novel 3-parameter generalized quaternion sequences of higher order Horadam numbers, which have not been studied before, are defined by investigating the relationship between generalized quaternions, which are important mathematical objects used in physics and mathematics, and higher order Horadam quaternions, which are extensions of the higher order Horadam numbers to quaternion algebra. Also, the recurrence relations of sequences whose members are higher order Horadam 3-parameter generalized quaternions are described. Furthermore, certain properties of these generalized quaternions are presented, such as the generating and exponential function, summation and Binet formula, and some identities resulting from these quaternions are obtained.