Linear functional equations

Question

Linear functional equations like $(x + 1)P (x) = (x − 10)P (x + 1)$ are fairly common in competition math, and there are some general techniques for proving things about the solutions, such as degree of polynomial solutions, highest or lowest degree coefficients, etc.

One could imagine a somewhat general theory for functional equations of the form $L f = g$, with $f$ the unknown, $g$ a given function, and $L$ a linear operator composed of the basic operations of multiplication by $x$, i.e. $f(x) \mapsto x f(x)$, and shifts $f(x) \mapsto f(x+a)$. Are there any published works on the theory of such equations?

Answer 1

You're talking about Difference Equations. But you could also interpret it as a Recurrence Equation or as a Finite Difference Equation. You could also interperate it as a $0$th-order Difference / Delay Differential Equation, or a delay equation (but that's probably less said).

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In Applications of the Symbolic Toolbox (Chapter $10.13$) by George Lindfield and John Penny they show a good method to solve such equations with constant coefficients. They use the unilateral $\mathcal{Z}$ transform with its usefull property of linearity and (where $\mathcal{Z}_{x}\left[ y\left( x \right) \right]\left( z \right) = Y\left( z \right)$ and $k \in \mathbb{Z}$): $$ \begin{align*} \mathcal{Z}_{x}\left[ y\left( x + k \right) \right]\left( z \right) = z^{k} \cdot Y\left( z \right) - \sum\limits_{m = 0}^{k - 1}\left[ z^{k - m} \cdot y\left( m \right) \right]\\ \mathcal{Z}_{x}\left[ y\left( x + k \right) \right]\left( z \right) = z^{-k} \cdot Y\left( z \right) + \sum\limits_{m = 1}^{k}\left[ z^{m - k} \cdot y\left( -m \right) \right]\\ \end{align*} $$

An example that they give is "$6 \cdot y\left( x \right) - 5 \cdot y\left( x - 1 \right) + y\left( x - 2 \right) = 4^{-x}$" and their solution is "$y\left( x \right) = \frac{5}{2} \cdot \left( \frac{1}{2} \right)^{x} - 2 \cdot \left( \frac{1}{3} \right)^{x} + \frac{1}{2} \cdot \left( \frac{1}{4} \right)^{x}$" wich is correct - see here for prove.

This method is also mentioned in TRANSFORM METHODS (Chapter $Z$-transforms) by S. Braun.

Another method to search for the homogeneous solution is reduction to the characteristic equation as discussed in Theory and Applications of Numerical Analysis (Chapter $13$ - ORDINARY DIFFERENTIAL EQUATIONS) by G.M. PHILLIPS and P.J. TAYLOR. You can find this method applied here too. More generally applied, you can find this method in Probability (Chapter $7$ - DIFFERENCE EQUATIONS).

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A work focused directly on polynomial solutions of difference equations is Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials by Olha Shkaravska and Marko van Eekelen (this article is even free to read):

This article addresses the problem of computing an upper bound of the degree $d$ of a polynomial solution $P\left( x \right)$ of an algebraic difference equation of the form $G\left( x \right) \left( P\left( x - \tau_{1} \right),\, \ldots,\, P\left( x - \tau_{s} \right) \right) + G_{0}\left( x \right) = 0$ when such $P\left( x \right)$ with the coefficients in a field $\mathbb{K}$ of characteristic zero exists and where $G$ is a non-linear $s$-variable polynomial with coefficients in $\mathbb{K}\left[ x \right]$ and $G_{0}$ is a polynomial with coefficients in $\mathbb{K}$.