Extension of Ss,n relations to negative indices

Prove that the relations Ss,n+1 = Ss+1,n + un x Ss+2,n−1 and Ss,n = Ss+1,n + un+s+1 x Ss+2,n, which are established for s ≥ 0 in Proposition 4.2, also hold for all integers s ≤ −1.

Background

Proposition 4.2 provides two identities connecting the formal power series Ss,n built from the discrete polynomials Sk(n) for nonnegative s. These identities underpin subsequent constructions involving Stieltjes continued fractions and Lax-type formal series. The authors conjecture that analogous relations should remain valid for negative s, but note that a proof is still required to substantiate this extension.