A Pair of Block-Alternating Infinite Products and Their Special Values

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.