# $f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$

Let f be defined from real to real

$f\left( x-1 \right) +f\left( x+1 \right) =\sqrt { 3 } f\left( x \right)$

Now how to find the period of this function f(x)? Can someone provide me a purely algebraic method to solve this problem please?

Update:My method

An elementary algebraic approach to the problem :

$f(x-1)+f(x+1)=\sqrt { 3 } f(x)$

Replace $x$ with $x+1$ and $x-1$ respectively.

We get $f(x)+f(x+2)=\sqrt { 3 } f(x+1)$ and $f(x-2)+f(x)=\sqrt { 3 } f(x-1)$

From these three equations we get $f(x-2)+f(x+2)=0$

Putting $x=x+2$ and adding with last equation we get $f(x-2)+f(x+4)=0$....(1)

Similarly $f(x-4)+f(x+2)=0$.....(2)

Put $x=x-6$ in (1)

We get $f(x-8)+f(x-2)=0$.....(3)

From (1) and (3) we get $f(x-8)=f(x+4)$

So the period of $f(x)$ is 12

• What is the period of $f(x)\equiv 0$? Commented May 9, 2015 at 22:42
• Isnt that a constant function ? 0 is the period.Why?
– user220382
Commented May 9, 2015 at 22:44
• Because this satisfies your condition. Commented May 9, 2015 at 22:44
• vadim123 is not mistaken, there is a trivial constant solution to your equation. $0+0=\sqrt{3} \cdot 0$. It is not necessary the only solution.
– Ian
Commented May 9, 2015 at 22:46
• Nice trick. I don't think I could have come up with it from scratch.
– Ian
Commented May 10, 2015 at 3:07

For each fixed $y \in [0,1)$, your equation is a second order linear recurrence relation with constant coefficients. This suggests that the first step should be to solve the characteristic equation $\lambda^2-\sqrt{3} \lambda + 1 = 0$. You find $\lambda_1,\lambda_2 = \frac{\sqrt{3} \pm i}{2}=e^{\pm i \pi/6}$. This means that, for $y \in [0,1)$ and $n \in \mathbb{Z}$:

$$f(y+n)=c_1 e^{i n \pi/6} + c_2 e^{-i n \pi/6}$$

where $c_1,c_2$ are specified by the choice of $f(y)$ and $f(y-1)$. This tells you that the period is no larger than $\frac{2 \pi}{\pi/6} = 12$. It can be smaller: for instance your equation has a trivial constant solution $f \equiv 0$.

• Presently I don't know recurrence.I would be happier with an algebraic solution.Let me know if you know any algebraic solution which can be obtained by replacing x with x+a or x-a in the equation.Anyway thanks.
– user220382
Commented May 9, 2015 at 23:19
• @SanchayanDutta One way or another you're going to have to find the form that I wrote, as far as I can tell. You might be able to do it by simply assuming a sinusoid form, but that's just doing what I wrote without as much motivation. The method I described comes about by finding exponential solutions, which in your case would satisfy $f(x+1)=cf(x)$ for some number $c$. You might perhaps rewrite the equation as $\cos(\pi/6) f(x) = \frac{f(x-1)+f(x+1)}{2}$ and get something out of that.
– Ian
Commented May 9, 2015 at 23:20
• Actually I want to learn your technique.Can you provide me some links so that I can learn recurrence from scratch?Btw I'm just in high school.Not even through with Precalculus.
– user220382
Commented May 10, 2015 at 2:31
• @SanchayanDutta This seems to be a pretty good treatment: www3.cs.stonybrook.edu/~rezaul/Fall-2012/CSE548/…
– Ian
Commented May 10, 2015 at 2:37

Algebraic (non-coanstant)? NO.
But, for example $$f(x) = \sin\frac{\pi x}{6}$$ is a solution.

• Sorry for the confusion I might have caused by "algebraic".I mean an algebraic technique rather than using recurrence.
– user220382
Commented May 10, 2015 at 2:16