Perturbation equations of a Proca field around a Kerr BH in the slow-rotation approximation up to second order. See Pani, Cardoso, Gualtieri, Berti and Ishibashi (2012) for details.

SetDirectory[NotebookDirectory[]];

Proca perturbation equations:

This loads the coefficients in the Appendix of the paper:

(*POLAR FUNCTIONS*)A0=(1/(2 (M^2) (r^6)))(E^(-I t ω)) ϵp ((1/(2 M-r))(J^2) ((ϵa)^2) ((12 (M^2)-2 (3+l+(l^2)) M r+l (1+l) (r^2)) u1[l][r]+r (I (10 M-r) r ω u2[l][r]-(2 M-r) (I r ω u3[l][r]+6 M (u1[l])'[r]+r (-I r ω (u2[l])'[r]+(2 M-r) (u1[l])''[r]))))+1/(l (1+l) (((-2 M+r))^2))(-l (1+l) (2 M-r) (2 l (2 M-r) r ((M^2) (r^2)-(J^2) ((ϵa)^2))+2 (l^2) (2 M-r) r ((M^2) (r^2)-(J^2) ((ϵa)^2))+2 (M^2) (2 M-r) (r^5) (μ^2)+(J^2) ((ϵa)^2) (-20 (M^2)-2 M r (-7+(r^2) (μ^2))+(r^2) (-2+2 (m^2)+(r^2) (μ^2)))-4 J m (M^2) (r^3) ϵa ω) u1[l][r]+r (2 I l (1+l) r (2 J m (M^2) (-2 M+r) ϵa+(M^2) (r^2) (8 (M^2)-6 M r+(r^2)) ω-(J^2) (6 (M^2)-7 M r+(r^2)) ((ϵa)^2) ω) u2[l][r]-(2 M-r) (I r ω (l (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+(l^2) (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+2 J m r ϵa (J m ϵa-2 (M^2) r ω)) u3[l][r]+2 M (2 M-r) (3 (J^2) l (1+l) ((ϵa)^2) (u1[l])'[r]+r (-I l (1+l) M (r^3) ω (u2[l])'[r]+2 I J m M ϵa (u3[l])'[r]+l (1+l) (2 (M^2) (r^2)-M (r^3)-2 (J^2) ((ϵa)^2)) (u1[l])''[r]))))));
A1=(1/(2 (M^2) (r^4)))(E^(-I t ω)) ϵp (-((I (4 J m (M^2) (-2 M+r) ϵa+2 (M^2) (2 M-r) (r^3) ω+(J^2) (4 (M^2)-4 M r+3 (r^2)) ((ϵa)^2) ω) u1[l][r])/(((-2 M+r))^2))-(((J^2) ((ϵa)^2) (2 I M (2 M-r) ω u1[l][r]+(l (-2 M+r)+(l^2) (-2 M+r)-2 M (r^2) (ω^2)) u2[l][r]+(2 M-r) (-2 I M r ω (u1[l])'[r]+(-2 M+r) (u3[l])'[r])))/(((-2 M+r))^2))+1/(l (1+l) (((2 M-r))^3))(l (1+l) r (2 l (2 M-r) ((M^2) (2 M-r) r+(J^2) ((ϵa)^2))+2 (l^2) (2 M-r) ((M^2) (2 M-r) r+(J^2) ((ϵa)^2))+8 J m (M^2) r (-2 M+r) ϵa ω+2 (M^2) (r^3) (-2 M+r) ((-2 M+r) (μ^2)-r (ω^2))+(J^2) ((ϵa)^2) ((2 M-r) (2 (m^2)+r (2 M+r) (μ^2))+r (((2 M+r))^2) (ω^2))) u2[l][r]+(2 M-r) (I l (1+l) r (4 J m (M^2) (-2 M+r) ϵa+2 (M^2) (2 M-r) (r^3) ω+(J^2) (4 (M^2)+(r^2)) ((ϵa)^2) ω) (u1[l])'[r]+(2 M-r) (l (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+(l^2) (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+2 J m r ϵa (J m ϵa-2 (M^2) r ω)) (u3[l])'[r])));
AL=(1/(M^2))((((J^2) ((ϵa)^2) (-2 I M r ω u1[l][r]+(2 M-r) (u2[l][r]-u3[l][r]+r (u2[l])'[r])))/(2 M-r))+1/(l (1+l) (((-2 M+r))^2))(-I l (1+l) r (4 J m (M^2) (-2 M+r) ϵa+2 (M^2) (2 M-r) (r^3) ω+(J^2) (4 (M^2)+(r^2)) ((ϵa)^2) ω) u1[l][r]+(2 M-r) (2 l (1+l) (2 M-r) ((M^2) (r^2)-(J^2) ((ϵa)^2)) u2[l][r]-(l (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+(l^2) (2 M-r) (2 (M^2) (r^2)-(J^2) ((ϵa)^2))+2 J m r ϵa (J m ϵa-2 (M^2) r ω)) u3[l][r]+2 l (1+l) (M^2) (2 M-r) (r^3) (u2[l])'[r])));
α0=(1/(2 l (1+l) (M^2) (r^4) (((-2 M+r))^2)))(E^(-I t ω)) (I l (1+l) r (4 J m (M^2) (-2 M+r) ϵa+2 (M^2) (2 M-r) (r^3) ω+(J^2) r (2 M+r) ((ϵa)^2) ω) u1[l][r]+l (1+l) (((-2 M+r))^2) (2 (M^2) (r^2)-3 (J^2) ((ϵa)^2)) u2[l][r]-8 (M^4) (r^4) (μ^2) u3[l][r]+8 (M^3) (r^5) (μ^2) u3[l][r]-2 (M^2) (r^6) (μ^2) u3[l][r]+8 J m (M^3) (r^2) ϵa ω u3[l][r]-4 J m (M^2) (r^3) ϵa ω u3[l][r]-4 (M^3) (r^5) (ω^2) u3[l][r]+2 (M^2) (r^6) (ω^2) u3[l][r]-4 (J^2) M (r^3) ((ϵa)^2) (ω^2) u3[l][r]-8 l (M^4) (r^3) (u2[l])'[r]-8 (l^2) (M^4) (r^3) (u2[l])'[r]+8 l (M^3) (r^4) (u2[l])'[r]+8 (l^2) (M^3) (r^4) (u2[l])'[r]-2 l (M^2) (r^5) (u2[l])'[r]-2 (l^2) (M^2) (r^5) (u2[l])'[r]+4 (J^2) l (M^2) r ((ϵa)^2) (u2[l])'[r]+4 (J^2) (l^2) (M^2) r ((ϵa)^2) (u2[l])'[r]-4 (J^2) l M (r^2) ((ϵa)^2) (u2[l])'[r]-4 (J^2) (l^2) M (r^2) ((ϵa)^2) (u2[l])'[r]+(J^2) l (r^3) ((ϵa)^2) (u2[l])'[r]+(J^2) (l^2) (r^3) ((ϵa)^2) (u2[l])'[r]+16 (M^5) (r^2) (u3[l])'[r]-16 (M^4) (r^3) (u3[l])'[r]+4 (M^3) (r^4) (u3[l])'[r]-48 (J^2) (M^3) ((ϵa)^2) (u3[l])'[r]+48 (J^2) (M^2) r ((ϵa)^2) (u3[l])'[r]-12 (J^2) M (r^2) ((ϵa)^2) (u3[l])'[r]-16 (M^5) (r^3) (u3[l])''[r]+24 (M^4) (r^4) (u3[l])''[r]-12 (M^3) (r^5) (u3[l])''[r]+2 (M^2) (r^6) (u3[l])''[r]+16 (J^2) (M^3) r ((ϵa)^2) (u3[l])''[r]-16 (J^2) (M^2) (r^2) ((ϵa)^2) (u3[l])''[r]+4 (J^2) M (r^3) ((ϵa)^2) (u3[l])''[r]);
ζ0=-(1/(l (1+l) (M^2) (2 M-r) (r^4)))(E^(-I t ω)) J ϵa (2 l (1+l) M (6 (M^2)-3 M r+J m r ϵa ω) u1[l][r]+I (-l (1+l) (J m (8 M-r) ϵa-6 (M^2) (r^2) ω) u2[l][r]+2 J m M (r^2) ϵa (ω^2) u3[l][r]+(2 M-r) (6 I l (1+l) (M^2) r (u1[l])'[r]+J m ϵa (l (1+l) r (u2[l])'[r]-8 M (u3[l])'[r]+(2 M-r) r (u3[l])''[r]))));
ρ0=(1/(l (1+l) (M^2) (2 M-r) (r^4)))(E^(-I t ω)) (J^2) ((ϵa)^2) (2 I l (1+l) M r ω u1[l][r]+3 l (1+l) (2 M-r) u2[l][r]-2 M (r^2) (ω^2) u3[l][r]-2 l M r (u2[l])'[r]-2 (l^2) M r (u2[l])'[r]+l (r^2) (u2[l])'[r]+(l^2) (r^2) (u2[l])'[r]+12 (M^2) (u3[l])'[r]-10 M r (u3[l])'[r]+2 (r^2) (u3[l])'[r]-4 (M^2) r (u3[l])''[r]+4 M (r^2) (u3[l])''[r]-(r^3) (u3[l])''[r]);
σ0=0;
Bt0=((4 (E^(-I t ω)) (J^2) ((ϵa)^2) ϵp (l (1+l) u1[l][r]+I r ω u3[l][r]))/(l (1+l) M (r^6)));
Bt1=-((2 (E^(-I t ω)) (J^2) ((ϵa)^2) ϵp (l (1+l) u2[l][r]+(2 M-r) (u3[l])'[r]))/(l (1+l) (M^2) (2 M-r) (r^4)));
BtL=0;
D0=(1/((M^2) (2 M-r) (r^6)))(E^(-I t ω)) (J^2) ((ϵa)^2) ϵp (-(12 (M^2)-2 (3+l+(l^2)) M r+l (1+l) (r^2)) u1[l][r]+r (-I (10 M-r) r ω u2[l][r]+(2 M-r) (I r ω u3[l][r]+6 M (u1[l])'[r]+r (-I r ω (u2[l])'[r]+(2 M-r) (u1[l])''[r]))));
D1=(((E^(-I t ω)) (J^2) ((ϵa)^2) ϵp (2 I M (2 M-r) ω u1[l][r]+(l (-2 M+r)+(l^2) (-2 M+r)-2 M (r^2) (ω^2)) u2[l][r]+(2 M-r) (-2 I M r ω (u1[l])'[r]+(-2 M+r) (u3[l])'[r])))/((M^2) (r^4) (((-2 M+r))^2)));
DL=((2 (J^2) ((ϵa)^2) (2 I M r ω u1[l][r]-(2 M-r) (u2[l][r]-u3[l][r]+r (u2[l])'[r])))/((M^2) (2 M-r)));
(*AXIAL FUNCTIONS*)
At0=((4 (E^(-I t ω)) J ϵa ϵp (l M+(l^2) M-J m ϵa ω) u4[l][r])/(l (1+l) M (r^5)));
At1=-((2 I (E^(-I t ω)) (J^2) m ((ϵa)^2) ϵp (u4[l])'[r])/(l (1+l) (M^2) (r^4)));
AtL=0;
B0=(((E^(-I t ω)) J ϵa ϵp (r ω (J m ϵa-2 (M^2) r ω) u4[l][r]+2 (M^2) (2 M-r) (u4[l])'[r]))/(l (1+l) (M^2) (2 M-r) (r^4)));
B1=((I (E^(-I t ω)) J ϵa ϵp (J m ϵa-2 (M^2) r ω) (u4[l])'[r])/(l (1+l) (M^2) (2 M-r) (r^3)));
BL=-((2 I J r ϵa (J m ϵa-2 (M^2) r ω) u4[l][r])/(l (1+l) (M^2) (2 M-r)));
β0=(1/(l (1+l) (M^2) (r^4) (((-2 M+r))^2)))(E^(-I t ω)) ((l (((-2 M+r))^2) ((M^2) (r^2)-(J^2) ((ϵa)^2))+(l^2) (((-2 M+r))^2) ((M^2) (r^2)-(J^2) ((ϵa)^2))+M (r^2) (4 (M^3) (r^2) (μ^2)+2 (J^2) r ((ϵa)^2) (ω^2)+M (2 J m r ϵa ω+(r^4) ((μ^2)-(ω^2)))+(M^2) (-4 J m ϵa ω+(r^3) (-4 (μ^2)+2 (ω^2))))) u4[l][r]+M (((-2 M+r))^2) ((-2 (M^2) (r^2)+6 (J^2) ((ϵa)^2)) (u4[l])'[r]-r (-2 (M^2) (r^2)+M (r^3)+2 (J^2) ((ϵa)^2)) (u4[l])''[r]));
η0=-((I (E^(-I t ω)) J ϵa (2 M (-l (1+l) M (r^2) ω+J m ϵa (l+(l^2)+(r^2) (ω^2))) u4[l][r]+J m (2 M-r) ϵa ((-6 M+2 r) (u4[l])'[r]+(2 M-r) r (u4[l])''[r])))/(l (1+l) (M^2) (2 M-r) (r^4)));
λ0=-((4 (E^(-I t ω)) (J^2) ((ϵa)^2) u4[l][r])/(M (r^5)));
γ0=(((E^(-I t ω)) (J^2) ((ϵa)^2) ((l (2 M-r)+(l^2) (2 M-r)+2 M (r^2) (ω^2)) u4[l][r]+(2 M-r) (-8 M (u4[l])'[r]+(2 M-r) r (u4[l])''[r])))/(l (1+l) (M^2) (2 M-r) (r^4)));

Definitions

(*SetDirectory[NotebookDirectory[]];
<<"coefficients.mx";*)
a=(ϵa(J/M));
Δ[r_]:=(r^2)+(a^2)-2M r;
f[r_]:=((Δ[r])/((r^2)+(a^2)));
SJ[X_]:=Normal[Series[X,{ϵa,0,2}]];
(***)

CHECK: each coefficient has defined parity:

Clear[u1,u2,u3,u4]
u1[l][r_]:=polar u1v[r];
u2[l][r_]:=polar u2v[r];
u3[l][r_]:=polar u3v[r];
u4[l][r_]:=axial u4v[r];
D[{A0,A1,AL,α0,ζ0,ρ0,σ0,Bt0,Bt1,BtL,D0,D1,DL},axial](*POLAR FUNCTIONS*)
D[{At0,At1,AtL,B0,B1,BL,β0,η0,λ0,γ0},polar](*AXIAL FUNCTIONS*)
Clear[u1,u2,u3,u4]

Rules:

rulel={l->l};
rulelp1={l->l+1};
rulelm1={l->l-1};
rulelp2={l->l+2};
rulelm2={l->l-2};

t-component

eq0l=(A0+ ((Q[l+1])^2)(D0+l Bt0)+((Q[l])^2)(D0-(l+1)Bt0))/.rulel;
eq0lp1=(Q[l+1](At0+l B0))/.rulelm1;
eq0lm1=(Q[l](At0-(l+1)B0))/.rulelp1;
eq0lp2=(Q[l+1]Q[l+2](D0+l Bt0))/.rulelm2;
eq0lm2=(Q[l]Q[l-1](D0-(l+1)Bt0))/.rulelp2;

EQf0=Simplify[eq0l+eq0lm1+eq0lp1+eq0lp2+eq0lm2];

Series[Simplify[EQf0((((1/(2 (M^2) (r^6)))(E^(-I t ω)) ϵp ))^(-1))],{ϵa,0,2}]//Simplify

r-component

eq1l=(A1+ ((Q[l+1])^2)(D1+l Bt1)+((Q[l])^2)(D1-(l+1)Bt1))/.rulel;
eq1lp1=(Q[l+1](At1+l B1))/.rulelm1;
eq1lm1=(Q[l](At1-(l+1)B1))/.rulelp1;
eq1lp2=(Q[l+1]Q[l+2](D1+l Bt1))/.rulelm2;
eq1lm2=(Q[l]Q[l-1](D1-(l+1)Bt1))/.rulelp2;

EQf1=Simplify[eq1l+eq1lm1+eq1lp1+eq1lp2+eq1lm2];

Series[Simplify[EQf1((((1/(2 (M^2) (r^4)))(E^(-I t ω)) ϵp ))^(-1))],{ϵa,0,2}]//Simplify

Lorenz-component (tortoise)

eqLl=(AL+ ((Q[l+1])^2)(DL+l BtL)+((Q[l])^2)(DL-(l+1)BtL))/.rulel;
eqLlp1=(Q[l+1](AtL+l BL))/.rulelm1;
eqLlm1=(Q[l](AtL-(l+1)BL))/.rulelp1;
eqLlp2=(Q[l+1]Q[l+2](DL+l BtL))/.rulelm2;
eqLlm2=(Q[l]Q[l-1](DL-(l+1)BtL))/.rulelp2;

EQfL=Simplify[eqLl+eqLlm1+eqLlp1+eqLlp2+eqLlm2];

Series[Simplify[EQfL((((1/(M^2))))^(-1))],{ϵa,0,2}]//Simplify

θ-component (tortoise)

eq3l=(l(l+1)α0-I m ζ0 +((Q[l+1])^2)l(l ρ0+σ0)+((Q[l])^2)(l+1)((l+1)ρ0-σ0))/.rulel;
eq3lm1=Q[l](-(l+1)(η0/.rulelm1)-I m((l-1)(γ0/.rulelm1)+(λ0/.rulelm1)));
eq3lp1=Q[l+1](l(η0/.rulelp1)+I m((l+2)(γ0/.rulelp1)-(λ0/.rulelp1)));
eq3lm2=Q[l-1]Q[l](l+1)(-(l-2)(ρ0/.rulelm2)-(σ0/.rulelm2));
eq3lp2=Q[l+2]Q[l+1]l(-(l+3)(ρ0/.rulelp2)+(σ0/.rulelp2));

EQf3=Simplify[eq3l+eq3lm1+eq3lp1+eq3lp2+eq3lm2];

Series[Simplify[EQf3((((1/(2 (M^2) (r^5)))(E^(-I t ω)) ))^(-1))],{ϵa,0,2}]//Simplify

φ-component (tortoise)

eq4l=(l(l+1)β0+I m η0 +((Q[l+1])^2)l(l γ0+λ0)+((Q[l])^2)(l+1)((l+1)γ0-λ0))/.rulel;
eq4lm1=Q[l](-(l+1)(ζ0/.rulelm1)+I m((l-1)(ρ0/.rulelm1)+(σ0/.rulelm1)));
eq4lp1=Q[l+1](l(ζ0/.rulelp1)-I m((l+2)(ρ0/.rulelp1)-(σ0/.rulelp1)));
eq4lm2=Q[l-1]Q[l](l+1)(-(l-2)(γ0/.rulelm2)-(λ0/.rulelm2));
eq4lp2=Q[l+2]Q[l+1]l(-(l+3)(γ0/.rulelp2)+(λ0/.rulelp2));

EQf4=Simplify[eq4l+eq4lm1+eq4lp1+eq4lp2+eq4lm2];

Series[Simplify[EQf4((((1/((M^2) (r^5)))(E^(-I t ω))  ))^(-1))],{ϵa,0,2}]//Simplify

Simplifying the equations:

This loads the equations computed above:

SetDirectory[NotebookDirectory[]];
<<"final_eqs.mx";
Clear[Δ,f];
a=(ϵa(J/M));
SJ[X_]:=Normal[Series[X,{ϵa,0,2}]];

eq1=EQf0;
eq2=EQf1;
eq3=EQf3;
eq4=EQf4;
eqL=EQfL;

Lorenz condition:

u1nl=Solve[eqL==0,u1[l][r]][[1]]//Simplify;
u1nlp=Solve[(eqL/.l->l+1)==0,u1[l+1][r]][[1]]//Simplify;
u1nlm=Solve[(eqL/.l->l-1)==0,u1[l-1][r]][[1]]//Simplify;
u1nlpp=Solve[(eqL/.l->l+2)==0,u1[l+2][r]][[1]]//Simplify;
u1nlmm=Solve[(eqL/.l->l-2)==0,u1[l-2][r]][[1]]//Simplify;
ruleu1=Union[u1nl,D[u1nl,r],D[u1nl,{r,2}]];
ruleu1lp=Union[u1nlp,D[u1nlp,r],D[u1nlp,{r,2}]];
ruleu1lm=Union[u1nlm,D[u1nlm,r],D[u1nlm,{r,2}]];
ruleu1lpp=Union[u1nlpp,D[u1nlpp,r],D[u1nlpp,{r,2}]];
ruleu1lmm=Union[u1nlmm,D[u1nlmm,r],D[u1nlmm,{r,2}]];
ruleu1V=ruleu1;

FINAL FORM:

syst=Solve[{SJ[eq2//.ruleu1]==0,SJ[eq3//.ruleu1]==0,SJ[eq4//.ruleu1]==0},{u2[l]''[r],u3[l]''[r],u4[l]''[r]}][[1]];
u2ppn2=SJ[u2[l]''[r]/.syst]//Simplify;
u3ppn2=SJ[u3[l]''[r]/.syst]//Simplify;
u4ppn2=SJ[u4[l]''[r]/.syst]//Simplify;

u2ppn2lp2=u2ppn2/.l->l+2;
u2ppn2lp1=u2ppn2/.l->l+1;
u2ppn2lm2=u2ppn2/.l->l-2;
u2ppn2lm1=u2ppn2/.l->l-1;
(*****)
u3ppn2lp2=u3ppn2/.l->l+2;
u3ppn2lp1=u3ppn2/.l->l+1;
u3ppn2lm2=u3ppn2/.l->l-2;
u3ppn2lm1=u3ppn2/.l->l-1;
(*****)
u4ppn2lp2=u4ppn2/.l->l+2;
u4ppn2lp1=u4ppn2/.l->l+1;
u4ppn2lm2=u4ppn2/.l->l-2;
u4ppn2lm1=u4ppn2/.l->l-1;

rulepm1={u2[l+1]''[r]->u2ppn2lp1,u2[l-1]''[r]->u2ppn2lm1,u3[l+1]''[r]->u3ppn2lp1,u3[l-1]''[r]->u3ppn2lm1,u4[l+1]''[r]->u4ppn2lp1,u4[l-1]''[r]->u4ppn2lm1};
rulepm2={u2[l+2]''[r]->u2ppn2lp2,u2[l-2]''[r]->u2ppn2lm2,u3[l+2]''[r]->u3ppn2lp2,u3[l-2]''[r]->u3ppn2lm2,u4[l+2]''[r]->u4ppn2lp2,u4[l-2]''[r]->u4ppn2lm2};
ruleu1TOT=Join[ruleu1,ruleu1lp,ruleu1lm,ruleu1lpp,ruleu1lmm];

u2ppn2bis=SJ[u2ppn2/.ruleu1TOT/.rulepm1/.rulepm2]//Simplify;
u3ppn2bis=SJ[u3ppn2/.ruleu1TOT/.rulepm1/.rulepm2]//Simplify;
u4ppn2bis=SJ[u4ppn2/.ruleu1TOT/.rulepm1/.rulepm2]//Simplify;

Three equations for u2[l], u3[l] and u4[l]:

u2[l]''[r]==u2ppn2bis

u3[l]''[r]==u3ppn2bis

u4[l]''[r]==u4ppn2bis

Continued fractions:

Slightly different notation than in the paper.
1)  σn --> τn.
2)  q--> -q. In particular, sign=1 for QNMs and sign=-1 for bound-state modes.

Axial

q=sign(Sqrt[(μ^2)-(ω^2)]);
α11[n_]:=(n+1)(n+1-4 I ω);
(*α11[n_]:=-4 (1+n) q^2 (1+n-4 I ω);*)
α12[n_]:=0;
α21[n_]:=0;
α22[n_]:=α11[n];
(***)
β11[n_]:=-2 (n^2)+(2/q)(4 I ω q+3 (q^2)-(ω^2)-q)n+(1/q)(3 (q^2)+(12 q-1)(ω^2)-4 (q^3)-q l(l+1)-12 I ω (q^2)+4 I ω q+4 I (ω^3));
β11[n_]:=(-q (1+l+l^2+2 n^2+n (2-6 q)+q (-3+4 q)-s^2)+4 I (1+2 n-3 q) q ω+(-1-2 n+12 q) ω^2+4 I ω^3)/q;
(*β11[n_]:=4 q (q (1+l+l^2+2 n^2+q (3+4 q)+n (2+6 q)-s^2)-4 I q (1+2 n+3 q) ω-(1+2 n+12 q) ω^2+4 I ω^3);*)
β12[n_]:=0;
β21[n_]:=0;
β22[n_]:=β11[n];
(***)
γ11[n_]:=(n^2)-(2/q)((q^2)-(ω^2)+2 I ω q)n-(1/(q^2))(-4 I ω (q^3)+4 I (ω^3)q+6 (ω^2)(q^2)-(q^4)-(ω^4)+(q^2));
γ11[n_]:=((q (n-q+s)-2 I q ω+ω^2) (n q-q (q+s)-2 I q ω+ω^2))/q^2;
(*γ11[n_]:=-4 (q (n+q-s)-2 I q ω-ω^2) (q (n+q+s)-2 I q ω-ω^2);*)
γ12[n_]:=0;
γ21[n_]:=0;
γ22[n_]:=γ11[n];
(******************************)
α[n_]:={{α11[n],α12[n]},{α21[n],α22[n]}};
β[n_]:={{β11[n],β12[n]},{β21[n],β22[n]}};
γ[n_]:={{γ11[n],γ12[n]},{γ21[n],γ22[n]}};

Polar

q=sign(Sqrt[(μ^2)-(ω^2)]);
α11[n_]:=I J m (1+n)+(1+n) (1+n-4 I ω);
α12[n_]:=0;
α21[n_]:=0;
α22[n_]:=I J m (1+n)+(1+n) (1+n-4 I ω);
(***)
β11[n_]:=(1/4) J m (2 I (1-4 n+6 q)-((3 n)/ω)-8 ω-((4 I (ω^2))/q))+((-q (l+(l^2)+2 (n^2)+n (2-6 q)+(-1+q) (1+4 q))+4 I (1+2 n-3 q) q ω+(-1-2 n+12 q) (ω^2)+4 I (ω^3))/q);
β12[n_]:=-1+((3 J m ω)/(l+(l^2)));
β21[n_]:=2l (l+1)+((3 J m (l+(l^2)+4 ((q^2)+(ω^2))))/(4 ω));
β22[n_]:=(1/4) J m (2 I (1-4 n+6 q)-((3 n)/ω)-8 ω-((4 I (ω^2))/q))+((-q (l+(l^2)+2 n (1+n)-3 (1+2 n) q+4 (q^2))+4 I (1+2 n-3 q) q ω+(-1-2 n+12 q) (ω^2)+4 I (ω^3))/q);
(***)
γ11[n_]:=((((q^2)-q (2+n-2 I ω)-(ω^2)) (q (2-n+q)+2 I q ω-(ω^2)))/(q^2))+((J m (-3 q (5-4 n+3 q)-4 I q (6-n+q) ω+(3+8 q) (ω^2)+4 I (ω^3)))/(4 q ω));
γ12[n_]:=3+J m ((3/(4 ω))-((3 ω)/(l+(l^2))));
γ21[n_]:=-((3 J m (3 l (1+l)+4 ((q^2)+(ω^2))))/(4 ω));
γ22[n_]:=((((q^2)-q (1+n-2 I ω)-(ω^2)) (q-n q+(q^2)+2 I q ω-(ω^2)))/(q^2))+((J m (-3 q (4-4 n+3 q)-4 I q (6-n+q) ω+(3+8 q) (ω^2)+4 I (ω^3)))/(4 q ω));
(***)
δ11[n_]:=((3 J m (7 (q^2)+3 q (5-2 n+4 I ω)-3 (ω^2)))/(4 q ω));
δ12[n_]:=-((9 J m)/(4 ω));
δ21[n_]:=((9 J l (1+l) m)/(4 ω));
δ22[n_]:=((3 J m (7 (q^2)-6 q (-2+n-2 I ω)-3 (ω^2)))/(4 q ω));
(***)
ρ11[n_]:=(3/4) J m (-8 I+((4 n-5 (3+q))/ω)+((3 ω)/q));
ρ12[n_]:=((9 J m)/(4 ω));
ρ21[n_]:=-((3 J l (1+l) m)/(4 ω));
ρ22[n_]:=(3/4) J m (-8 I+((-12+4 n-5 q)/ω)+((3 ω)/q));
(***)
τ11[n_]:=((3 J m (q (5-n+q)+2 I q ω-(ω^2)))/(4 q ω));
τ12[n_]:=-((3 J m)/(4 ω));
τ21[n_]:=0;
τ22[n_]:=((3 J m (q (4-n+q)+2 I q ω-(ω^2)))/(4 q ω));
(******************************)
α[n_]:={{α11[n],α12[n]},{α21[n],α22[n]}};
β[n_]:={{β11[n],β12[n]},{β21[n],β22[n]}};
γ[n_]:={{γ11[n],γ12[n]},{γ21[n],γ22[n]}};
δ[n_]:={{δ11[n],δ12[n]},{δ21[n],δ22[n]}};
ρ[n_]:={{ρ11[n],ρ12[n]},{ρ21[n],ρ22[n]}};
τ[n_]:={{τ11[n],τ12[n]},{τ21[n],τ22[n]}};

References:

Pani, Cardoso, Gualtieri, Berti, Ishibashi: “Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric” (2012)