The Complex-Type k -Pell Numbers and Their Applications

In this study, a new sequence called the complex-type $k$ -Pell number is defined. Also, we give properties of this sequence such as the generating matrix, the generating function, the combinatorial representations, the exponential representation, the sums, the permanental and determinantal representations, and the Binet formula. Then, we determine the periods of the recurrence sequence according to the modulo $\upsilon$ and produce cyclic groups with the help of the generating matrices of the sequence. We also get some findings about the ranks and periods of the complex-type $k$ -Pell sequence. Additionally, we create relations between the orders of the cyclic groups produced and the periods of the sequence. Then, this sequence is moved to groups and examined in detail in finite groups. As an application, we get the periods of the complex-type $2$ -Pell numbers in the polyhedral groups $\left(\upsilon ,\mathrm{2,2}\right)$ , $\left(2,\upsilon ,2\right)$ , and $\left(\mathrm{2,2},\upsilon \right)$ and the quaternion group $Q_{2^{\upsilon }}$ .