On the Rational Recursive Sequence

On the Rational Recursive Sequence X_{N+1}=ƔX_{N-K}+(AX_N+BX_{N-K})⁄(CX_N-DX_{N-K})

Limit distribution of the degrees in scaled attachment random recursive trees

We study the limiting distribution of the degree of a given node in a scaled attachment random recursive tree, a generalized random recursive tree, which is introduced by Devroye et. al (2011). In a scaled attachment random recursive tree, every node $i$ is attached to the node labeled $lfloor iX_i floor$ where $X_0$, $ldots$ , $X_n$ is a sequence of i.i.d. random variables, with support in [0,...

On a Simultaneous Approximation Problem concerning Binary Recurrences

Let Rn (n = 0, 1, 2, . . .) be a second order linear recursive sequence of rational integers defined by Rn = ARn−1+BRn−2 for n > 1, where A and B are integers and the initial terms are R0 = 0, R1 = 1. It is known, that if α, β are the roots of the equation x2 −Ax−B = 0 and |α| > |β|, then Rn+1/Rn −→ α as n −→ ∞. Approximating α with the rational number Rn+1/Rn, it was shown that ∣∣∣α− Rn+1 Rn ∣...

On a (2,2)-rational Recursive Sequence

We investigate the asymptotic behavior of the recursive difference equation yn+1 = (α+ βyn)/(1 + yn−1) when the parameters α < 0 and β ∈ R. In particular, we establish the boundedness and the global stability of solutions for different ranges of the parameters α and β. We also give a summary of results and open questions on the more general recursive sequences yn+1 = (a+ byn)/(A+Byn−1), when th...

On the Rational Recursive Sequence xn + 1 = (A + sum k i = 0 alpha i xn - i)/(B + sum k i = 0 beta i xn - i)

On the rational recursive sequence yn = A + yn-1/yn-m for small A

This work studies the existence of positive prime periodic solutions of higher order for rational recursive equations of the form yn = A + yn−1 yn−m , n = 0, 1, 2, . . . , with y−m , y−m+1, . . . , y−1 ∈ (0,∞) and m ∈ {2, 3, 4, . . .}. In particular, we show that for sufficiently small A > 0, there exist periodic solutions with prime period 2m +Um + 1, for almost all m, where Um = max{i ∈ N : i...