Closed formula for the Adams-operation polynomials p_n

Derive a closed formula valid for all integers n ≥ 1 for the monic polynomials p_n(x) in Z_ε[τ, γ^{±1}, x]/⟨τ^2−2hγ, (1+ε)τ⟩ that satisfy p_1(x)=1, p_2(x)=x+2τ and the recursion p_n(x)=(x+τ)p_{n−1}(x)−γ p_{n−2}(x)+ψ^{n−1}(τ), and that compute ψ^n(x)/x for the unstable Adams operations on hermitian K-theory; these polynomials control the Todd class associated to the Adams operations.

Background

In the computation of Todd classes for Adams operations on hermitian K-theory (KO), the authors introduce monic polynomials p_n(x) defined by a specific recursion that encode ψ^n(x)/x, where ψ^n denotes the unstable Adams operations.

While initial cases can be computed inductively, a general closed-form expression would streamline Todd class computations and deepen understanding of the algebraic structure of Adams operations in the quadratic setting.