Some identities deriving from the nth power of a special matrix

In this paper, we consider the Horadam sequence and some summation formulas involving the terms of the Horadam sequence. We derive combinatorial identities by using the trace, the determinant and the n th power of a special matrix

On bicomplex fibonacci numbers and their generalization

In this chapter, we consider bicomplex numbers with coefficients from Fibonacci sequence and give some identities. Moreover, we demonstrate the accuracy of such identities by taking advantage of idempotent representations of the bicomplex numbers. And then by this representation, we give some identities containing these numbers. We then make a generalization that includes these new numbers and we call them Horadam bicomplex numbers. Moreover, we obtain the Binet formula and generating function of Horadam bicomplex numbers for the first time. We also obtain two important identities that relate the matrix theory to the second order recurrence relations