Lucas sequences: where do $U_n$ and $V_n$ come from?

Question

Reading the Wikipedia page, I’m trying to understand where the two Lucas sequences $U_n(P, Q)$ and $V_n(P, Q)$ come from. The page starts off with

$$f_n=Pf_{n-1}-Qf_{n-2}$$

The characteristic equation is easy enough to find (my personal favorite way).

$$x^2-Px+Q=0$$

The roots of this equation are: $$\alpha = \frac{P+\sqrt{D}}{2} \quad \beta = \frac{P - \sqrt{D}}{2}, \qquad \text{where } D = P^2 - 4Q$$

Thus any form of $C_1\alpha^n+C_2\beta^n$ satisfy the recurrence. What's so special about $U_n(P, Q)$ and $V_n(P, Q)$? The page also annoyingly says "one quickly verifies" the following:

$$\alpha^n = \frac{V_n + U_n \sqrt{D}}{2}, \quad \beta^n = \frac{V_n - U_n \sqrt{D}}{2}$$

But still both of those can be substituted out for other variants. However, it seems that these two definitions allow for a bunch of important things.

What I’m confused about is:

1. How do $U_n$ and $V_n$ arise naturally from the recurrence relation?
2. Why are their initial conditions typically taken as $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, and $V_1 = P$?
3. Can they be derived directly without first assuming their existence?

In short, I’d like to understand the logic or motivation behind the construction of $U_n$ and $V_n$ and how they connect to the recurrence beyond being “predefined” sequences.

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Edit: I would like to revisit @CyeWaldman's interesting point about representing $f_n$ in the following form: $$f_n=f_0\color{blue}{S_n}+f_1\color{blue}{T_n}\tag{1}$$ From calculations, it turns out $\color{blue}{T_n}=U_n$; somehow, I initially miscalculated $\color{blue}{S_n}=V_n$, accepted the answer, and proceeded to find properties of $U_n$ and $V_n$ given by $\rm eq. (1)$. However, it turned out $\color{blue}{S_n}$ actually was $$\color{blue}{S_n}=\frac{\alpha\beta^n-\alpha^n\beta}{\alpha-\beta}$$ Does the representation $\rm eq. (1)$ give straightforward properties that makes $U_n$ and $V_n$ interesting? Is $S_n$ important or does it have interesting properties/identities?

Answer 1

I find the Wikipedia reference to be somewhat strange, insofar as $a^n,b^n$ are defined i.t.o. $U_n,V_n$ and $U_n,V_n$ are defined i.t.o. $a^n,b^n$. This is circular and of no help.

However, I can explain what you looking for in terms of a general second-order linear recurrence as follows. I have written frequently on these pages of the general solution to the sequence $f_n=af_{n-1}+bf_{n-2}$, with arbitrary initial conditions, $f_{0,1}$. See, for example, here.

The general solution can be expressed as

$$ f_n=\left(f_1-\frac{af_0}{2}\right) \frac{\alpha^n-\beta^n}{\alpha-\beta}+\frac{f_0}{2} (\alpha^n+\beta^n) $$

where $\alpha,\beta=(a\pm\sqrt{a^2+4b})/2$.

This can also be expressed as

$$ f_n=\left(f_1-\frac{af_0}{2}\right)F_n+\frac{af_0}{2}L_n $$

where

$$ F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},\quad L_n=\frac{\alpha^n+\beta^n}{\alpha+\beta} $$

I call these the Fibonacci- and Lucas-type terms because they give the sequences of those names when $a,b=1$ and the appropriate initial conditions are specified.

Ergo,

$$ (\alpha-\beta)F_n=\alpha^n-\beta^n\\ (\alpha+\beta)L_n=\alpha^n+\beta^n $$

Adding and subtracting these equations we find that

$$ \alpha^n=\bigg(\frac{(\alpha-\beta)F_n+(\alpha+\beta)L_n}{2}\bigg)\\ \beta^n=\bigg(\frac{(\alpha-\beta)F_n-(\alpha+\beta)L_n}{2}\bigg) $$

I think you should be able to take it from here. The relation between $F_n,L_n$ and $U_n,V_n$ should be clear.