# Relationship between a linear difference equation and the hyperbolic functions [closed]

Considering a linear difference equation $$\begin{equation} \underbrace{\begin{bmatrix} -p & 1 & 0 & 0 & 0 & \cdots & 0\\ 1 &-p & 1 & 0 & 0 & \cdots & 0\\ 0 & 1 &-p & 1 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots &\vdots\\ \vdots & \vdots & \vdots &\vdots & \ddots & \vdots &\vdots\\ 0 & \cdots & 0 & 0 & 1 & -p & 1 \\ 0 & \cdots & 0 & 0 & 0 & 1 & -p \end{bmatrix}}_P \underbrace{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ \vdots \\ x_{N-2} \\ x_{N-1} \end{bmatrix}}_x = -\underbrace{\begin{bmatrix} x_0 \\ 0 \\ 0 \\ \vdots \\ \vdots \\ 0 \\ 0 \end{bmatrix}}_C, \end{equation}$$ where $$p:=2+\tilde{\kappa}^2 \tau^2$$ and $$\tau:=T/N$$ are given; sequence $$\{x_j\}_{j=1}^{N-1}$$ needs to be estimated by solving the linear difference equation.

Normally, the solution of a well-defined linear difference equation can be estimated using linear least squares (LLS), such that $$x=-P^{-1}C$$; however, according to the paper "Optimal Execution of Portfolio Transactions":

"a linear difference equation whose solution may be written as a combination of the exponentials $$e^{\pm \kappa(j\tau)}$$, where $$\kappa$$ satisfies $$\begin{equation} \frac{2}{\tau^2}(\cosh(\kappa\tau)-1) = \tilde{\kappa}^2 \end{equation}$$ and the solution can be written as $$\begin{equation} x_j = \frac{\sinh(\kappa(T-j\tau))}{\sinh(\kappa T)} x_0. \end{equation}$$"

Can anyone explain to me how $$P^{-1}$$ is related to the hyperbolic function $$\sinh$$?

• I’m voting to close this question because it is more suited for math SE but to old to migrate – kjetil b halvorsen Jan 30 at 21:43