Factorizations and representations of second order linear recurrences with indices in arithmetic progressions
Let \(\{V_n\}\) and \(\{U_n\}\) be generalized Lucas and Fibonacci sequences satisfying a general second order linear recurrence relation with Lucas and Fibonacci type initial conditions. The authors find second order linear recurrences for the sequences \(\{V_{\pm kn}\}\) and \(\{U_{\pm kn}\}\). Using these recurrence relations, the authors show that the members of these sequences are the determinants of certain tridiagonal matrices. Further, the authors present representations and trigonometric factorizations of the members of these sequences applying matrix methods and methods relying on Chebyshev polynomials of the first and second kinds. The authors also give the generating functions and some combinatorial representations of these sequences.
second order linear recurrence sequences
scientific article; zbMATH DE number 5917660