On 3-parameter Generalized Quaternions with Higher Order Lichtenberg Numbers Components

In this study, we introduce a new class of 3-parameter generalized quaternions, whose components are defined using higher order Lichtenberg numbers. These structures, called higher order Lichtenberg 3-parameter generalized quaternions, are investigated from algebraic and analytical perspectives. The study aims to extend the role of number sequences in quaternion algebra and to contribute to the development of new mathematical tools. Several basic properties of this quaternion family are established, including recurrence relations and Binet-type formulas that provide explicit representations of the terms. Moreover, generating functions and exponential generating functions are derived, offering compact forms and useful analytical frameworks. In addition, a number of fundamental identities are presented to illustrate the structural features of the quaternions under consideration. Finally, the results obtained here not only generalize well-known concepts but also suggest possible applications of higher order Lichtenberg quaternions in combinatorics, coding theory, and theoretical physics.