Discrete Mathematics

Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

10.2: The 2nd-order linear homogeneous recurrence relation with constant coefficiens

Definition

Let k ∈Z⁺and C₀,C₁, ...,C_k be real numbers. If{an}is a discrete function, then

C₀a_n+C₁a_n−1+C₂a_n−2+ · · · +C_ka_n−k =f(n), n ≥k, is a linear recurrence relation with constant coefficiens of order k. When f(n) =0 for all n≥0, the relation is called homogeneous; otherwise, it is called nonhomogeneous.

Consider the homogeneous relation of order 2:

C₀a_n+C₁a_n−1+C₂a_n−2=0, n≥2.

Definition

The quadratic equation

C₀r²+C₁r +C₂=0 is called the characteristic equation.

Let r₁,r₂denote the (characteristic) roots of the equation.

Consider the homogeneous relation of order 2:

C₀a_n+C₁a_n−1+C₂a_n−2=0, n≥2.

Definition

The quadratic equation

C₀r²+C₁r +C₂=0 is called the characteristic equation.

Let r₁,r₂denote the (characteristic) roots of the equation.

Consider the homogeneous relation of order 2:

C₀a_n+C₁a_n−1+C₂a_n−2=0, n≥2.

Definition

The quadratic equation

C₀r²+C₁r +C₂=0 is called the characteristic equation.

Let r₁,r₂denote the (characteristic) roots of the equation.

Case A: r₁and r₂are distinct real roots.

a_n=c₁r₁ⁿ+c₂r₂ⁿ.

Example (the Fibonacci relation) Solve the recurrence relation

F_n+2=F_n+1+Fn, n≥0, and F₀=0, F₁=1.

Case A: r₁and r₂are distinct real roots.

a_n=c₁r₁ⁿ+c₂r₂ⁿ.

Example (the Fibonacci relation) Solve the recurrence relation

F_n+2=F_n+1+Fn, n≥0, and F₀=0, F₁=1.

Case A: r₁and r₂are distinct real roots.

a_n=c₁r₁ⁿ+c₂r₂ⁿ.

Example (the Fibonacci relation) Solve the recurrence relation

F_n+2=F_n+1+Fn, n≥0, and F₀=0, F₁=1.

Case B: r₁ and r₂are complex roots.

Example

Solve the recurrence relation a_n=2(an−1−a_n−2), where n ≥2 and a₀=1, a₁=2.

Case B: r₁ and r₂are complex roots.

Example

Solve the recurrence relation a_n=2(an−1−a_n−2), where n ≥2 and a₀=1, a₁=2.

Example

Let b ∈R⁺, and D_n denote the n×n determinant given by 

b b 0 0 0 ... 0 0 0 0 0 b b b 0 0 ... 0 0 0 0 0 0 0 b b b ... 0 0 0 0 0 . . . ... . . . 0 0 0 0 0 ... b b b 0 0 0 0 0 0 0 ... 0 b b b 0 0 0 0 0 0 ... 0 0 b b b 0 0 0 0 0 ... 0 0 0 b b

Find the value of D_n as a function of n.

Case C: Repeated real roots.

Example

Solve the recurrence relation a_n+2=4a_n+1−4a_n, where n≥0 and a₀=1, a₁=3.

Theorem

Consider the homogeneous relation of order 2:

C₀a_n+C₁a_n−1+C₂a_n−2=0, n≥2.

Let r denote the characteristic root of multiplicity 2, then the general solution has the form

(A0+A₁n)rⁿ where A₀and A₁are arbitrary constants.

Thank you.