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

Deromediqm 2022-07-21 Answered
Finding polynomials sattisfying P ( c + K / ( u + c ) ) ( u + c ) 2 / K = P ( u )
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Answers (1)

esbalatzaj
Answered 2022-07-22 Author has 15 answers
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
{ A = 1 K ( A c 2 B c + C ) B = 2 c K ( A c 2 B c + C ) 2 A c + B C = c 2 K ( A c 2 B c + C ) + c ( 2 A c + B )
Solving it, we obtain that P ( u ) B ( u + c ) or a degenerated solution K=0 and P(−c)=0.
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