The Generation of Higher-order Linear Recurrences from Second-order Linear Recurrences
نویسنده
چکیده
Along the lines of this theorem, Selmer [1] has shown how one can form a higher-order linear recurrence consisting of the term-wise products of two other linear recurrences. In particular, let {sn} be an m-order and {tn} be a p-order linear integral recurrence with the associated polynomials s(x) and t(x), respectively. Let a^, i = 1,2, ..., m, and 3j, j = 1, 2, ..., p, be the roots of the polynomials s(x)andt(x), respectively, and assume that each polynomial has no repeated roots. Then, the sequence
منابع مشابه
Generating Matrices for Weighted Sums of Second Order Linear Recurrences
In this paper, we give fourth order recurrences and generating matrices for the weighted sums of second order recurrences. We derive by matrix methods some new explicit formulas and combinatorial representations, and some relationships between the permanents of certain superdiagonal matrices and these sums.
متن کاملFactorizations 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 b...
متن کاملFactorizations and representations of the backward second-order linear recurrences
We show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second-order linear recurrences.Also considering how to the negatively subscripted terms of second-order linear recurrences can be connected to Chebyshev polynomials by determinants of these matrices, we give factorizations and representations of these number...
متن کاملSecond-order Linear Recurrences of Composite Numbers
In a well-known result, Ronald Graham found a Fibonacci-like sequence whose two initial terms are relatively prime and which consists only of composite integers. We generalize this result to nondegenerate second-order recurrences.
متن کاملSecond-order bounds for linear recurrences with negative coefficients
This paper introduces a generalization of Fibonacci and Pell polynomials in order to obtain optimal second-order bounds for general linear recurrences with negative coefficients. An important aspect of the derived bounds is that they are applicable and easily computable. The results imply bounds on all entries in inverses of triangular matrices as well as on coefficients of reciprocals of power...
متن کامل