Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products
- 1Tianjin College, University of Science and Technology Beijing, Tianjin 301830, China
- 2School of Science, Tianjin University of Technology and Education, Tianjin 300222, China
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.
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BibTeX
@article{liang2026grammatical,
title = {{Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products}},
author = {Dengxing Liang and Harold Ruilong Yang and Yanjun Zhang},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203690},
url = {https://www.mdpi.com/2227-7390/14/20/3690},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Grammatical Constructions for Binomial Convolutions of Fibonacci-Type Numbers and Their Products
AU - Dengxing Liang
AU - Harold Ruilong Yang
AU - Yanjun Zhang
PY - 2026
DA - 2026/10/11
JO - Mathematics
DO - 10.3390/math14203690
UR - https://www.mdpi.com/2227-7390/14/20/3690
AB - 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.
SN - 2227-7390
ER -