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} =…
Cauchy functional equation
f(x+y) = f(x) + f(y). Continuous solutions: f(x) = cx. Without continuity: pathological solutions exist (need axiom-of-choice). …
Abel's functional equation
α(f(x)) = α(x) + 1. Linearises iteration of f via 'Abel-coordinate' α. Solution exists near hyperbolic fixed points. Used in dynamical…
Fibonacci recurrence (golden ratio)
F_n+1 = F_n + F_n-1 with F_0 = 0, F_1 = 1. Closed-form Binet F_n = (φ^n - ψ^n)/√5 where φ = (1+√5)/2 golden ratio. Limiting ratio…
Characteristic equation for linear recurrences
Linear recurrence Σ c_k a_{n-k} = 0 has solutions a_n = r^n where r solves polynomial c_0 r^k + c_1 r^{k-1} + ... + c_k = 0. Repeated…
Z-transform (discrete signals)
Z{x[n]} = Σ x[n] z^{-n} maps discrete-time sequences to complex z-plane functions. Discrete analogue of Laplace transform. …
Schröder functional equation (iteration)
σ(f(x)) = λ σ(x). Linearises f near hyperbolic fixed-point with multiplier λ. Composition f^n closed-form via σ. Used for analytic…
Z-transform (Jury 1958)
Jury 1958 unilateral Z-transform Z{x_n} = sum x_n z^-n; difference-equation x_{n+1}=ax_n+bu_n -> X(z); discrete-time signal-processing…
Logistic map (Feigenbaum 1975)
Feigenbaum 1975 x_{n+1}=r x_n(1-x_n); period-doubling at r=3, 3.449,...; Feigenbaum constants delta=4.6692 + alpha=2.5029 universal in…
Jury stability test (1962)
Jury 1962 discrete-time stability: characteristic polynomial p(z) has all roots inside |z|<1 iff Jury table conditions on coefficients…
Linear difference eqn (characteristic roots)
Solution of a_k x_{n-k}+...+a_0 x_n=0 is x_n = sum c_i lambda_i^n with lambda_i roots of characteristic polynomial; Fibonacci satisfies…
Euler-Maclaurin summation (1735-1742)
Euler 1735 + Maclaurin 1742: sum_{n=a}^b f(n) = integral f + (f(a)+f(b))/2 + sum B_{2k}/(2k)! [f^{(2k-1)}]; Bernoulli-number remainder;…
Cauchy functional equation (1821)
Cauchy 1821: f(x+y)=f(x)+f(y); continuous solutions are f(x)=cx; without continuity (assuming AC) Hamel-basis pathological solutions exist…
Recurrence (Fibonacci 1202)
L Fibonacci 1202 Liber Abaci; modern Binet-formula + golden-ratio + matrix-form via diagonalization.
Pringsheim radius (1894)
A Pringsheim 1894 power-series radius-of-convergence; modern complex-analytic Z-transform-region of convergence.
Functional equations (Aczel 1966)
J Aczel 1966 'Lectures on Functional Equations'; modern Bailey + Castillo + Pales survey treatments.
Dynamical systems (Poincare 1890)
H Poincare 1890 three-body-problem; foundational text in dynamical-systems + KAM 1962 + Smale 1967.
Lorenz attractor (1963)
E Lorenz 1963 deterministic-non-periodic-flow; foundational text in chaos + butterfly-effect.
Hahn q-difference (1949)
W Hahn 1949 q-difference + q-orthogonal-polynomials; modern q-deformed quantum-groups + non-commutative geometry.