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

Source 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.