Functional Equation Season II

Algebra Level 3

Is there any periodic function with fundamental period less than 100 that satisfies the equation

f ( x + 1 ) + f ( x − 1 ) = 2 ( cos ⁡ π 2018 ) f ( x ) ? f(x+1)+f(x-1)=2(\cos{\frac{\pi}{2018}}) f(x)?

Inspiration .

No Yes

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Arturo Presa by Arturo Presa , 62, USA

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1 solution

Arturo Presa
Jun 13, 2018

The answer is yes. An example of such a function can be f ( x ) = sin ⁡ ( 4037 2018 π x ) . f(x)=\sin{(\frac{4037}{2018}\pi x)}. It easy to see that the period of this function is 4036 4037 , \frac{4036}{4037}, that is less than 1, and we can also prove that it satisfies the given functional equation. Indeed, using the identity sin ⁡ A + sin ⁡ B = 2 sin ⁡ ( A + B 2 ) cos ⁡ ( A − B 2 ) , \sin {A}+\sin{ B}= 2\sin{(\frac{A+B}{2})}\cos{(\frac{A-B}{2})}, we obtain that

f ( x + 1 ) + f ( x − 1 ) = sin ⁡ ( 4037 2018 π ( x + 1 ) ) + sin ⁡ ( 4037 2018 π ( x − 1 ) ) = 2 sin ⁡ ( 4037 2018 π x ) cos ⁡ ( 4037 2018 π ) = 2 cos ⁡ π 2018 f ( x ) f(x+1)+f(x-1)= \sin{(\frac{4037}{2018}\pi (x+1))}+\sin{(\frac{4037}{2018}\pi (x-1))}=2\sin{(\frac{4037}{2018}\pi x)}\cos{(\frac{4037}{2018}\pi )}=2\cos{\frac{\pi}{2018}} f(x)

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