The generating function of a family of the sequences in terms of the continuant

Let a0, a1, … , ar−1 be positive numbers and define a sequence {qm}, with initial conditions q0 = 0 and q1 = 1, and for all m ⩾ 2, qm = atqm−1 + qm−2 where m ≡ t(mod r). For r = 2, the author called the sequence {qm} as the generalized Fibonacci sequences and studied it in [1]. But, it remains open to find a closed form of the generating function for general {qm}. In this paper, we solve this open problem, that is, we find a closed form of the generating function for {qm}in terms of the continuant.