True or False? Homogeneous linear recurrence equations are linear combinations of power functions.
I think the answer is false because although a homogeneous linear recurrence equation is a linear combination, it is composed of constant coefficients. However to solve this equation the solution may include powers... so I'm second guessing myself. Would that make this statement True?
Apparently I was not the only one who was confused if this was True or False? This was the professor's response:
"Nice! #6 is not counted, because question is somewhat ambiguous: Homogeneous linear recurrence relations are not linear combinations of power functions. However the solutions to homogeneous linear recurrence relations are linear combinations of powers of the roots of the characteristic equations."
The statement is false. Homogeneous linear recurrence equations are not generally linear combinations of power functions.
To understand why, let's first clarify what a homogeneous linear recurrence equation is. It is an equation that relates a sequence to its previous terms using a linear combination. For example, a typical homogeneous linear recurrence equation is of the form:
a[n] = c₁a[n-1] + c₂a[n-2] + ... + cₖa[n-k]
where a[n] represents the nth term of the sequence and c₁, c₂, ..., cₖ are constant coefficients.
In general, the solution to a homogeneous linear recurrence equation consists of a linear combination of powers of the roots of the characteristic equation associated with the recurrence equation. The characteristic equation is obtained by assuming a solution of the form a[n] = rⁿ, where r is some unknown constant.
The characteristic equation typically has the form:
rₖ - c₁rₖ₋₁ - c₂rₖ₋₂ - ... - cₖ = 0
The solutions of this characteristic equation, say r₁, r₂, ..., rₖ, can be real or complex numbers. The general solution to the homogeneous linear recurrence equation can then be expressed as a linear combination of the corresponding powers of the roots:
a[n] = A₁r₁ⁿ + A₂r₂ⁿ + ... + Aₖrₖⁿ
where A₁, A₂, ..., Aₖ are constants determined by any initial conditions.
Therefore, while the solution to a homogeneous linear recurrence equation may involve powers of the roots, the equation itself is not a linear combination of power functions.