Conjecture on expressing H-coefficients via invariants for equation (5.3)

Prove that for any integer g ≥ 1 and for solutions of the recurrence Un+g Sg+1(n) = Un+g+1 Sg+1(n+2) (equation (5.3)), the coefficients H_{g+k+1} satisfy H_{g+k+1} = (−1)^{g+k} 2·9_{g,k} for k = 0,…,g, and that for all k ≥ 2g+2 the coefficients H_k are polynomials in (H1,…,Hg; 9_{g,0},…,9_{g,g}).

Background

The coefficients Hk arise from the series y(x)=1+Σ{j≥1} Hj x^j satisfying y^2=f(x), and are birationally related to the invariants ck of the Volterra map. Computations for g=1 and g=2 suggest direct expressions of higher Hk in terms of the invariants 9g,r that appear in the polynomial recurrences built from the discrete polynomials Sk(n). This motivates Conjecture 5.15 connecting H{g+k+1} to 9_{g,k} and asserting polynomial dependence of the remaining Hk on (H1,…,Hg) and (9_{g,0},…,9_{g,g}).