We investigate the solutions of certain meta-Fibonacci recurrences form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets initial conditions. When $f(n)=1$ $n\leq 1$, we prove that resulting integer sequence is closely related to function c

Febrile seizures (FS) occur in 2–5% of children. Their pathogenesis is unknown. Elevated levels of prostaglandins (PG) have been found in cerebrospinal fluid after such seizures, and a third of all patients have recurrences. No safe ways of reducing the risk of recurrences have been found. The outcome has been shown in prospective studies to be good, by they have been linked to mesial temporal ...

In recent work on multivariate elliptic hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of t...

We consider an initially bound quantum particle subject to an external time-dependent field. When the external field is large, the particle shows a tendency to repeatedly return to its initial state, irrespective of whether the frequency of the field is sufficient for escape from the well. These recurrences, which are absent in a classical calculation, arise from the system evolving primarily l...

We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an = an−1 + n−d k=d f(n, k)akan−k where, very roughly speaking, f(n, k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlevé I equations are discussed in detail. 1 Main results There are many examples in the ...

We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the method is demonstrated by its application to several dynamical systems, including the Kuramoto-Sivashinsky system.

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.

Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th cyclic resultant of f is rm = Res(f, x m − 1). A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree d is determined by its first 2d+1 cy...