Concepts

• Recurrence
• Function f(n) defined for n = 0, 1, 2, ... (or some other initial segment of integers)
• Equation or inequality that describes f(n) in terms of f(k), k < n
• Examples

f(n) = f(n-1) + f(n-2) // Fibanacci 
g(n) = 2g(n/2) + n     // integer arithmetic

• Initial conditions
• Required values for first few inputs
• Examples:

f(1) = f(0) = 0 // Fibanacci
may be unspecified

• Exact Solution
• Find (set of all) function(s) f(n) satisfying the recurrence (and the ICs)
• Example: Fibanacci
  f(n) = (φ₁ⁿ - φ₂ⁿ) ⁄ √5
• Asymptotic Solution
• Example:
  g(n) ≤ O( n log n )

Linear Recurrences, Polynomial Method

• Linear recurrence of order k: f(n) = a₁f(n - 1) + a₂f(n - 2) + ... + a_kf(n - k), where the coefficients a_k are constants
• Characteristic polynomial: P(x) = x^k - a₁x^k-1 - a₂x^k-2 - ... - a_k
• Characteristic equation: P(x) = 0 , or x^k = a₁x^k-1 + a₂x^k-2 + ... + a_k
• Roots of characteristic polynomial are solutions to recurrence
• If r₁ and r₂ are roots then f(n) = c₁r₁ⁿ + c₂r₂ⁿ is a solution for any constants c₁ and c₂ .
• Degree two case:
• f(n) = a₁f(n - 1) + a₂f(n - 2)
• Suppose r² - a₁r - a₂ has two distinct roots r₁ and r₂.
  Then the sequence {f_n} is a solution of the recurrence iff
  f_n = c₁r₁ⁿ + c₂r₂ⁿ for n = 0, 1, ..., where c₁ and c₂ are constants.
• Suppose r² - a₁r - a₂ has only one root r₀. Then the sequence {f_n} is a solution of the recurrence iff
  f_n = c₁r₀ⁿ + c₂nr₀ⁿ for n = 0, 1, ..., where c₁ and c₂ are constants.
• General degree n case - mutatis mutandis
• Example: Fibanacci