Factorizations and Representations of Second Order Linear Recurrences with Indices in Arithmetic Progressions

In this paper we consider second order recurrences {Vk} and {Un} . We give second order linear recurrences for the sequences {V±kn} and {U±kn}. Using these recurrence relations, we derive relationships between the determinants of certain matrices and these sequences. Further, as generalizations of the earlier results, we give representations and trigonometric factorizations of these sequences by matrix methods and methods relying on Chebyshev polynomials of the first and second kinds. We give the generating functions and some combinatorial representations of these sequences.