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Triangular system (5.20) via a general identity for Q and T polynomials

Prove the identity Σ_{j=0}^k (-1)^j Q_{k−j}(n+j) T_{s+k−j}(n) = 0 for all integers k ≥ 1 and s ≥ k, and thereby derive the triangular linear system (5.20) for the invariants g(k,s) in full generality.

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Background

Invariants for the two-parameter class of recurrences Un+k Sk(n) = Un+s Sk(n+2) are constructed using discrete polynomials Qk(n) and G(k,s)(n). The authors propose that a set of invariants satisfies a triangular linear system (equation (5.20)), which can be established if a specific identity involving Q and T polynomials holds for general k and s. While this identity is verified for k=1 trivially and for k=2,3 in prior work [19], a general proof is currently unavailable.

References

It should be noted that, in principle, we could prove (5.20) if we prove the identity >(−1)jQk−j(n+j)T3+k−j(n) =0 for any k ≥ 1 and s ≥ k. For k = 1 this identity is trivial, while for two cases k = 2 and k = 3 it has been proved in the work [19]. However, it should be noted that the lack of proof of this statement does not prevent us from verifying this fact for any specific case.

— Volterra map and related recurrences (2502.06908 - Svinin, 10 Feb 2025) in Section 5.4, Remark 5.12