Remove["Global`*"]

Coefficients for the inverse elliptic integral of the first kind :

(Subscript[a, F])[n_]:=(1/(2n+1))Sum[Binomial[n,n/2+i]((Gamma[n/2+i+1/2])/(Gamma[1/2]))((Gamma[n/2-i+1/2])/(Gamma[1/2]))(m^(n/2-i)),{i,-n/2,n/2}]
(Subscript[b, F])[n_,l_]:=(((-1))^l)((Gamma[2n+1+l])/(Gamma[2n+1]))

Expansion terms for the inverse elliptic integral of the first kind :

Table[Simplify[((u^(2n+1))/((2n+1)n!))Sum[(Subscript[b, F])[n,l]BellY[n,l,Table[(Subscript[a, F])[i],{i,1,n-l+1}]],{l,1,n}]],{n,1,4}]

Coefficients for the inverse elliptic integral of the second kind :

(Subscript[a, E])[n_]:=(1/(2n+1))Sum[Binomial[n,n/2+i]((Gamma[n/2+i+1/2])/(Gamma[1/2]))((Gamma[n/2-i-1/2])/(Gamma[-1/2]))(m^(n/2-i)),{i,-n/2,n/2}]
(Subscript[b, E])[n_,l_]:=(((-1))^l)((Gamma[2n+1+l])/(Gamma[2n+1]))

Expansion terms for the inverse elliptic integral of the second kind :

Table[Simplify[((u^(2n+1))/((2n+1)n!))Sum[(Subscript[b, E])[n,l]BellY[n,l,Table[(Subscript[a, E])[i],{i,1,n-l+1}]],{l,1,n}]],{n,1,4}]