Nested recursions with ceiling function solutions

作者：Isgur Abraham; Kuznetsov Vitaly; Tanny Stephen M^*

来源：Journal of Difference Equations and Applications, 2012, 18(6): 1015-1026.

DOI：10.1080/10236198.2010.540573

• nested recursion
• Conolly sequence
• Conway sequence
• ceiling function

Consider a nested, non-homogeneous recursion R(n) defined by
R(n) = Sigma R-k(i=1)(h - s(i) - Sigma R-pi(j=1)(n - q(ij))) + nu,
with c initial conditions R(1) =xi(1) > 0, R(2) = xi(2) > 0, ... , R(c) = xi(c) > 0, where the parameters are integers satisfying k > 0, p(i) > 0, and a(ij) > 0. We develop an algorithm to answer the following question: for an arbitrary rational number r/q, is there any set of values for k; pi; si; aij and n such that the ceiling function inverted right perpendicularrn/qeinverted left perpendicular is the unique solution generated by R(n) with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to R(n). The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form inverted right perpendicularrn/qeinverted left perpendicular is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if inverted right perpendicularrn/qeinverted left perpendicular is the solution generated by any recursion R(n) of the form above, then r = 1. We also use our ceiling function methodology to derive the first known connection between the recursion R(n) and a natural generalization of Conway's recursion.

• 出版日期2012