Block-Alternating Wallis-Type Products and Non-Commuting Boundary Limits

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22831291
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

Block-Alternating Wallis-Type Products and Non-Commuting Boundary Limits

Masanori Fujii
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

Block-Alternating Wallis-Type Products and Non-Commuting Boundary Limits

Masanori Fujii
preprint en

Abstract

We begin with a Wallis-type infinite product of ratios of differences of squares. Collecting its even- and odd-indexed factors gives the classical Wallis product Wₑ = π/2 and its odd-indexed counterpart Wₒ = 4/π. Applying a block-alternating sign rule to the factors, we introduce a two-parameter family of infinite products A(K,d) and B(K,d). The product A(K,d) admits a complete closed form in terms of real gamma values: in particular, A(K,1) is a rational multiple of π², while A(K,2) is a rational multiple of G² for odd K and of π²G² for even K, where G = 1/AGM(1,√2) = Γ(1/4)² / (2^(3/2) π^(3/2)) is the Gauss constant. The appearance of G is tied to the singular value 1/√2 of the elliptic modulus, which enters at d = 2. The companion B(K,d) is defined by replacing differences of squares by sums. A boundary relation connects the two families: A(∞,1) = π/2 and lim (K→∞) log B(K,∞) = −π/2, so the Wallis limit π/2 reappears with opposite sign in the logarithm of B. This latter value is an iterated limit (d→∞ before K→∞), while lim (K→∞) B(K,d) = 1 for every finite d. The two iterated limits therefore disagree, so the two-variable limit of B(K,d) at (∞,∞) does not exist. Version 2. Substantially revised and retitled (formerly "A Pair of Block-Alternating Infinite Products and Their Special Values"). Reorganized around a non-commuting boundary-limit theorem for B(K,d) (the "Wallis symmetry"); the Gauss-constant expression is corrected; a general closed form, a universal recurrence, and a scaling identity are added.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
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