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Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products

Year: 2026
Type: article
Abstract: In 2023, Yang introduced the Fibonacci grammar to study convolutions of Fibonacci and Lucas numbers. We develop this framework by incorporating the inverse operator D−1 and a sign-parametrized convolution grammar. These operators give grammatical expressions for Fibonacci-type numbers Tn(a,b) at arbitrary integral indices and lead to the Lucas dual Sn=Tn(2b−ap,bp+2a). We obtain unified addition and subtraction identities for this dual pair and a single representation covering ordinary and alternating binomial convolutions of both sequences. As an application, we correct the parity condition in a lemma of Kılıç, Ömür and Türker Ulutaş, which is valid only for odd scaling factors, and determine which of its consequences survive. We then construct a four-letter product grammar for the binomial convolutions of two generalized Fibonacci or Lucas numbers. Its two-channel form reorganizes the corresponding root evaluations and makes the parity dependence explicit. The same decomposition gives an arbitrary-weight formula; a scaling reduction treats positive second parameters, while a direct four-root form treats the real, distinct, nonzero-root Horadam regime. Finally, we extend the grammar to finitely many factors, prove the even-step parity criterion, and give the triple UUU formula as an explicit example. The genuinely additional structural results are the inverse-operator construction, the channel decomposition, the even-step criterion, and the triple-product formula; the overlapping closed evaluations are identified as equivalent reformulations of earlier root-based formulas.
Source: Mathematics
DOI: 10.3390/math14203690
Language: English
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7221510126
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Cites: 14
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Related to: 10