Positivity of second order linear recurrent sequences

Positivity of second order linear recurrent sequences

We give an elementary proof for the Positivity Problem for second order recurrent sequences: it is decidable whether or not a recurrent sequence defined by un = aun−1 + bun−2 has only nonnegative terms.

Reciprocal Sums of Second Order Recurrent Sequences

Let Z and IR (C) denote the ring of the integers and the field of real (complex) numbers, respectively. For a field F, we put i* = F\{0}. Fix A GC and B G C*, and let X(A, B) consist of all those second-order recurrent sequences {wn}neZ of complex numbers satisfying the recursion: n+i = „ ~ n-i (i.e., Bwn_Y = Awn wn+l) for n = 0, ± 1, ± 2,.... (1) For sequences in SE(A,B), the corresponding cha...

Intersections of Second - Order Linear Recursive Sequences

Positivity Problems for Low-Order Linear Recurrence Sequences

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomi...

Recurrent metrics in the geometry of second order differential equations

Given a pair (semispray $S$, metric $g$) on a tangent bundle, the family of nonlinear connections $N$ such that $g$ is recurrent with respect to $(S, N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N, g)$ to be recurrent as well as for the triple $(S, stackrel{c}{N}, g)$ where $stackrel{c}{N}$ is the canonical ...