We introduce a pair of infinite products defined by quadratic rational factors together with a block-alternating sign rule governed by a floor function. The resulting two-parameter family of products A (K, d) and B (K, d) exhibits several notable special values and boundary limits. In particular, A (1, 1) = π²/8, A (1, 2) = 2G², and A (1, 3) = 3/2, where G = Γ (1/4) ² / (2π^ (3/2) ) ≈ 0. 8346 denotes the Gauss constant. A particularly simple boundary behavior occurs when the block parameter tends to infinity: A (K, ∞) = 2K. In the opposite direction, letting K → ∞ yields limits expressible in terms of gamma-function ratios, including the classical value A (∞, 1) = π/2. The companion product B (K, d), obtained from sum-of-squares factors, forms a natural dual structure and generates exponential constants such as B (∞, ∞) = e^ (-π/2) and B (1, ∞) = 2π / sinh (π). A structurally distinguished fixed point is A (1/2, d) = B (1/2, d) = 1 for all d > 0.
Masanori Fujii (2026) studied this question.