For each retained t, set E_t(T)=product_{q!=0,t} ((T-q)/(t-q))^(q_q), ignoring zero multiplicities. E_t(t)=1. Form R_t(z) as the first q_t coefficients of F(t+alpha z)/E_t(t+alpha z). Then F(0)=sum_t E_t(0) R_t(-t/alpha). To see why, replace zero by T in this formula. The term indexed by t has the required jet there; all other terms vanish to the required order. The resulting polynomial has degree below sum q_t, so Hermite uniqueness identifies it with F. All divisions here are over K; finite-precision validity still needs a separate bound.