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I'm solving a problem to which the final obstacle (I think) is knowing if this function is even or odd:

Given $$xQ(x+2018)=(x-2018)Q(x)$$ and that $Q(1)=1$, what is $Q(2017)$?

My best attempt so far is plugging in $x =-1$ to which I got $$(-1)Q(-1+2018)=(-1-2018)Q(-1)$$ $$Q(2017)=(2019)Q(-1)$$ I can simply conclude $Q(2017)=2019$ if I can prove $Q(-1)=Q(1)$ which brings me to the question how can I prove that $xQ(x+2018)=(x-2018)Q(x)$ is an even function? I've only been used to identifying odd and even functions for the usual polynomial/rational functions with exponents and all.

I would really appreciate the help. Thank you!

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  • $\begingroup$ @robgimarino: As currently stated, there's not enough information. Are you sure you've stated the problem fully? $\endgroup$ – quasi Nov 2 '18 at 5:46
  • $\begingroup$ I've stated the problem exactly as reflected in the paper $\endgroup$ – robgimarino Nov 2 '18 at 6:59
  • $\begingroup$ What paper? A homework sheet? Can you post an image or a link? $\endgroup$ – quasi Nov 2 '18 at 7:21
  • $\begingroup$ @Robgimarino I can disprove your claim. Try to find f(-2017) which you will get 2017/4035. If f(x)=f(-x) for all x,then f(-2017)=f(2017) proving that your claim f(1)=f(-1) is wrong. $\endgroup$ – Jasmine Nov 3 '18 at 7:45
  • $\begingroup$ I must be looking at the problem wrongly, I see. Thanks for letting me know :) I'll try to look at it in another way $\endgroup$ – robgimarino Nov 3 '18 at 9:53

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