x)$?

Asked Dec 24 '21 at 16:59

Active Dec 24 '21 at 17:08

Viewed 137 times

So this is actually a part of another question:

If a polynomial $f$ satisfies: $$\log_2[f(x)]=\log_2\left[2+\frac 23+\frac 29+\cdots +\infty\right]\cdot \log_3 \left[1+ \frac{f(x)}{f(1/x)} \right] $$ if $f(10)=1001,$ find $f(20)$

So, I actually thought, to find $f(x)$ polynomial altogether, since I couldn't get any another clue. simplifiying that long equation yields us this equation : $$f(x)+f(1/x)=f(x)f(1/x)$$, after which i am stuck and unable to proceed further. Putting different values of $x$ won't benefit, as far I tried. Any help is massively appreciated and thanks in advance.

edited Dec 24 '21 at 17:08

lulu

70,402

asked Dec 24 '21 at 16:59

Kshitij Kumar

I don't remember the derivation at the moment, but the answer to such functional equation is $f(x)=1±x^n$, which in this case should be $f(x)=1+x^3$. This might help, I guess. – ultralegend5385 Dec 24 '21 at 17:02
@ultralegend5385, Thank you for the answer, but I came here specificly for the derived answer. Thanks anyways. – Kshitij Kumar Dec 24 '21 at 17:13
Can you elaborate on the series of fractions? We have $2+2/3+2/9+\cdots$, a decreasing sequence, but it ends in $\infty$... What's the pattern? – FShrike Dec 24 '21 at 17:18
@FShrike Its an infinite GP with common ratio $1/3$ – Kshitij Kumar Dec 24 '21 at 17:19
A very simple solution to the functional equation is $f(x) = 1+x$. – DOUGLAS BRUNSON Dec 24 '21 at 17:31
1

Does this answer your question? $f(x)f(1/x)=f(x)+f(1/x)$ See also this and this. All found via approach0. – Jakob Streipel Dec 24 '21 at 17:34
Actually, this is asked really very very often. – Jakob Streipel Dec 24 '21 at 17:40

How to find $f(x)$, if $f(x)+f(1/x)=f(x)f(1/x)$?

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