A q-weighted analogue of the Trollope-Delange formula

The paper’s Theorem 1 states exactly the identity 1/n·Sq(n) = (q/2)((1 − q^[log2 n]+1)/(1 − q) − q^[log2 n] F̂q(log2 n)) with F̂q(u) = 2^{1−u} T_a(2^{−(1−u)}), a=1/(2q) for |q|>1/2, q≠1, and proves it via functional equations for Gq and a de Rham system leading to Fq(x)=qx−(1/2)Ta(x) and then to the claimed formula (formulas (26)–(28) in the paper; see Theorem 1 and its proof). This matches the statement and derivation in the PDF precisely . The model’s solution derives the same identity by direct bit-counting: it computes Ni(n), rewrites via the tent function τ, obtains the finite dyadic sum (29), and then identifies the Ta-series at a dyadic argument (using that the series truncates at dyadic rationals) to reach the same closed form, consistent with the paper’s representation (29) and with the definition of Ta and its dyadic truncation property . The model also flags a minor parenthesization issue in the prompt (not in the paper); the paper’s formula uses (1 − q^{k+1})/(1 − q), which is correct .