Two closely related branches. Difference equations study sequences (x_n) satisfying a recursion F(x_{n+k}, x_{n+k-1}, …, x_n, n) = 0 — the discrete-time analogue of ODEs. Linear-constant-coefficient theory is complete: given F(x_{n+k},…
difference-and-functional-equations
Linear recurrence a_{n+k} + c_{k-1}a_{n+k-1} + ... = 0: characteristic-poly solution
Constant-coefficient linear recurrence of order k: a_{n+k} + c_{k-1}a_{n+k-1} + … + c_0 a_n = 0. Ansatz a_n = λⁿ reduces the recurrence to…
Cauchy/Jensen/d'Alembert functional equations: regularity → linearity
Three classical functional equations. Cauchy's additive equation f(x + y) = f(x) + f(y), solved by any linear f(x) = c x under any one of:…
Z-transform X(z) = Σ_n x[n] z^{-n}: discrete-time analogue of Laplace
The (unilateral) Z-transform X(z) = Σ_{n=0}^∞ x[n] z^{-n} maps a causal sequence (x[n])_{n≥0} to a complex function X(z) on a region |z| >…
Binet F_n=(φⁿ−ψⁿ)/√5: F₅=5, F₁₀=55; recurrence residual 0
Sympy-exact closed-form verification of Binet's 1843 formula for the Fibonacci recurrence F_{n+2} = F_{n+1} + F_n with F_0 = 0, F_1 = 1. …
Cauchy additive f(x+y)=f(x)+f(y), f(1)=2 ⇒ f(x)=2x; f(3)=6
Sympy-exact witness that under the Q-homogeneity ansatz f(x) = c x, Cauchy's additive functional equation f(x + y) = f(x) + f(y) is…
Z{u[n]} = z/(z−1) on |z|>1; Z(2) = 2; Z(3) = 3/2
Sympy-exact closed form for the Z-transform of the unit step. The unit step u[n] = 1 (n ≥ 0) has Z-transform X(z) = Σ_{n≥0} 1 · z^{-n} =…