First of all, we will be using the following theorem proven in class about the recurrence relation of |s(n, k)| = S * (n, k). holds for any n, k, and defines a recurrence relation for S * (n, k). Let us define (t) n = t(t + 1)(t + 2) · · · (t + n − 1) to be the polynomial in t with roots 0, −1, −2,. .. , n − 1. Let us now define a(n, k) as follows (t) n = n k=0 a(n, k)t k. (1) We want to find a...

We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first pos...

A compartmental carbon model is presented in which there are parameters quantifying the transfer between various carbon pools. These parameters are estimated using two CO2 data sets (elevated and ambient) from the Duke Forest FACE data set. A quadratic objective criterion is used to compare closeness of fit of system output CO2 flux to the observed CO2 flux data. Adjoint equations are developed...

We show that the known matrix representations of the stationary state algebra of the Asymmetric Simple Exclusion Process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the constant term method (CTM) and bijective combinatorial methods to express many forms of the ASEP normali-sation factor in terms of Ballot numbers. One partic...

In this paper, we investigate the global behavior of the difference equation of order three xn+1 = α+ xn−2 xn , n = 0, 1, . . . where the parameters α, k ∈ (0,∞) and the initial values x−2, x−1 and x0 are arbitrary positive real numbers.

In this paper, existence criteria for single and multiple positive solutions of periodic boundary value problems for first order difference equations of the form

and Applied Analysis 3 After k steps we obtain the following formula xn A ⎛ ⎝ A x r/p n−k ⎛ ⎝ A x r/p2 n−k x r/p n−k−1 ⎛ ⎝ A x r/p3 n−k x r/p2 n−k−1x r/p n−k−2