Pensieve header: Converting symmetric rational functions of $\omega$ to rational functions of $v$.

a=((1-3 ω+5 (ω^2)-5 (ω^3)+5 (ω^4)-3 (ω^5)+(ω^6))/(ω (1-2 ω+(ω^2)-2 (ω^3)+(ω^4))))

Simplify[(a/.ω->(ω^(-1)))==a]

a/.ω->x+I y

FullSimplify[a/.ω->x+I y,{ x∈Reals,y∈Reals}]

FullSimplify[a/.ω->x+I (((1-(x^2)))^(1/2))]

Factor[a/.ω->x+I y]

Reduce[a==y∧x==ω+(ω^(-1)),x]

Simplify[a/.ω->(x-(ω^(-1)))]

First@Solve[x==ω+(ω^(-1)),ω]

a/.First@Solve[x==ω+(ω^(-1)),ω]

FullSimplify[a/.First@Solve[x==ω+(ω^(-1)),ω]]//Together//ExpandNumerator//ExpandDenominator

Expand[NumeratorDenominator[a]/(ω^3)]

PolynomialQuotient[Numerator[a]/(ω^3),1+(ω^2),ω]

PolynomialRemainder[Numerator[a]/(ω^3),1+(ω^2),ω]

PolynomialRemainder[Numerator[a]/(ω^3),((1+(ω^2))/ω),ω]

Denominator[-5+(1/(ω^3))-(3/(ω^2))+(5/ω)+5 ω-3 (ω^2)+(ω^3)]

ω2[v_][p_]:=Module[{q=Expand[p],n,c},
If[q===0,0,c=Coefficient[q,ω,n=Exponent[q,ω]];c (v^n)+ω2v[q-c(((ω+(ω^(-1))))^n),v]]];

ω2[v][-5+(1/(ω^3))-(3/(ω^2))+(5/ω)+5 ω-3 (ω^2)+(ω^3)]

f=((1-3 ω+5 (ω^2)-5 (ω^3)+5 (ω^4)-3 (ω^5)+(ω^6))/(ω (1-2 ω+(ω^2)-2 (ω^3)+(ω^4))))

{num,den}=NumeratorDenominator[f]

{num,den}/=(ω^(Exponent[num,ω]/2))

Times@@(ω2[v]/@{num,den})

Times@@(ω2[v]/@{num,den})/.v->(2(u^2)-1)/2

Factor[Times@@(ω2[v]/@{num,den})/.v->(2(u^2)-1)/2]

ω2u[f_]:=Factor@Module[{num,den,v},
{num,den}=NumeratorDenominator[f];{num,den}/=(ω^(Exponent[num,ω]/2));
Times@@(ω2[v]/@{num,den})/.v->(2(u^2)-1)/2 ]