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A family of restricted subadditive recursions

By Roger J. Wallace


AbstractFor a fixed non-negative integer k, let Uk={Uk(n)}, n⩾0, denote that sequence which is defined by the initial conditions Uk(0)=Uk(1)=Uk(2)=⋯=Uk(k)=1, and by the restricted subadditive recursionUk(n+k+1)=min0⩽l⩽[k/2](Uk(n+l)+Uk(n+k−l)),n⩾0.Uk is important in the theory of optimal sequential search for simple real zeros of real valued continuous kth derivatives. The structure of Uk depends substantially on the parity of k. In an earlier work, the author proved that U2p (p a fixed non-negative integer) also satisfies a certain periodic system of p+1 difference equations. This system was solved, and several closed form expressions for U2p(n), n>2p, were duly exhibited. In contrast, much less is known about the behaviour of U2p+1, although it has been conjectured that it satisfies, eventually, a single (solvable) difference equation. In this paper, the author determines a sufficient condition for U2p+1 to satisfy this equation. It transpires that this finding on U2p+1 is a special case of a general conclusion on members of a certain family of restricted subadditive recursions

Publisher: Elsevier Science B.V.
Year: 2002
DOI identifier: 10.1016/S0166-218X(01)00335-3
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