2
votes
0answers
52 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
0
votes
1answer
50 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
3
votes
2answers
208 views

Recurrence relations on a continuous domain

While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, ...
0
votes
1answer
41 views

In which cases iterative equations can be reduced to finite-difference equations?

In which cases iterative equations can be reduced to finite-difference equations and when they can't?
3
votes
2answers
220 views

Understanding difference equation

I was given an example $$R_n = R_{n-1} + R_{n-2} $$ This equation is given as an second-order equation. Why is it so?