with(LinearAlgebra):
applygauge := proc(trec,trel)  # input rec in terms of _u and _n and relation in form v(_n)=a1(_n)_u(_n+1) + a0(_n)_u(_n)
local rel, rec, a, b, s, ShouldBeZero, temp1, temp2, temp3, d;
rel := eval(trel, {_c[1]=1, _c[2]=1});
rec := trec;
s := v(_n+2) + a(_n)*v(_n+1) + b(_n)*v(_n);
ShouldBeZero := eval(s, {rel, subs(_n=_n+1, rel), subs(_n=_n+2, rel)}); 
temp1 := solve(rec, _u(_n+2));
temp2 := solve( eval(rec, _n=_n+1), _u(_n+3));
temp3 := subs(_u(_n+3)=temp2, _u(_n+2) = temp1, ShouldBeZero);
temp3 := collect(temp3, {_u(_n), _u(_n+1)}, factor);
ShouldBeZero := { coeffs(temp3, {_u(_n), _u(_n+1)}) };
d := solve(ShouldBeZero, {a(_n), b(_n)});
subs(d, s);
end: