Finding polynomials sattisfying P ( − c + K / ( u + c ) ) ( u + c ) 2 / K = P ( u )

Question

Finding polynomials sattisfying   P            (        −  c  +  K      /    (  u  +  c  )            )        (  u  +  c      )    2        /    K  =  P  (  u  )

esbalatzaj · Accepted Answer

When u tends to infinity then the modulus of the left side of the equality (*) grows not faster than a polynomial of degree 2. Therefore the polynomial from the right side of the equality has a degree at most two. Now, assuming that

P (x) = A x^{2} + B x + C

and comparing coefficients from both sides, we obtain a system

{\begin{cases} A = \frac{1}{K} (A c^{2} - B c + C) \\ B = \frac{2 c}{K} (A c^{2} - B c + C) - 2 A c + B \\ C = \frac{c^{2}}{K} (A c^{2} - B c + C) + c (- 2 A c + B) \end{cases}

Solving it, we obtain that

P (u) \equiv B (u + c)

or a degenerated solution K=0 and P(−c)=0.

Finding polynomials sattisfying P(−c+K/(u+c))(u+c)2/K=P(u)

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