As shown in the title, I want to show that if a rational function satisfies h ( t ) = h ( 1 / t ), then h(t) can be express in the variable 1 + 1 / t, that is, h ( t ) = f ( t + 1 / t ) for some rational function f. Any help is apreciated.

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As shown in the title, I want to show that if a rational function satisfies   h  (  t  )  =  h  (  1      /    t  ), then h(t) can be express in the variable   1  +  1      /    t, that is,   h  (  t  )  =  f  (  t  +  1      /    t  ) for some rational function   f. Any help is apreciated.

hospitaliapbury · Accepted Answer

Let   S denote the set of images of       Q    ∖  {  0  } under   t  ↦  t  +  1      /    t. For   x  ∈  S, the equation   t  +  1      /    t  =  x is equivalent to       t    2    −  x  t  +  1  =  0, whose roots are of the form   u  ,    1      /    u for   u  ∈      Q    ∖  {  0  }. Define   g  (  x  ) as the root of minimum modulus, so   g  :  S  ↦  (      Q    ∖  {  0  }  )  ∩  [  −  1  ,    1  ] and  t  =  g  (  x  )    ⟹    x  =  t  +  1      /    t  .Finally, define   f  (  x  )  :=  h  (  g  (  x  )  ), so  t  =  g  (  x  )    ⟹    h  (  t  )  =  f  (  x  )  =  f  (  t  +  1      /    t  )  .For each   t  ∈      Q    ∖  {  0  }, the above calculation works for some   x  ∈  S.

As shown in the title, I want to show that if a rational function satisfies h ( t )

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