Some properties of Frobenius–Sigmoid–Fibonacci polynomials with their applications
Abstract
In this paper, we introduce two-variable Frobenius–Sigmoid–Fibonacci polynomials and
their associated numbers within the approach of Golden F-Calculus. Utilizing generating
functions, we derive several fundamental properties, including summation theorems, recurrence
relations, symmetry properties, and F-derivative identities. We further establish connections
with Stirling–Fibonacci numbers of the second kind and present multiple summation
formulas and convolution–type identities. Additionally, we introduce novel parametric forms
of these polynomials, incorporating trigonometric generating structures, and analyze their behavior
using the F-differential operator and functional equation techniques. Furthermore, we
derive a formula that elucidates the relationship between this matrix and the generalized Pascal
matrix through the application of Fibonomial coefficients of the first kind. The proposed
approach enriches the theory of Fibonacci–based special polynomials and opens new avenues
for applications in combinatorics, number theory, approximation theory, and matrix analysis.
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References
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\bibitem{Kanan1} M. Elbes, T. Kanan, M. Alia and M. Ziad, COVD-19 detection platform from X-ray images using deep learning, International Journal of Advances in Soft Computing and its Applications, 14 (1), pp. 197-211, 2022.
\bibitem{Kanan2} T. Kanan, M. Elbes, K. Abu Maria, M. Alia, Exploring the potential of IoT-based learning environments in education, International Journal of Advances in Soft Computing and its Applications, 15 (2), 2023.
\bibitem{Baitha} I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh and Sh. Momani, Design fractional-order PID controllers for single-joint robot arm model, International Journal of Advances in Soft Computing and its Applications, 14(2), 2022, pp. 96-114.
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\bibitem{BOR} A. Z. Border, The $r$--Stirling numbers, \textit{ Disc. Math.,} \textbf{49} (1984), 241--259.
\bibitem{9} H. K. Kwan, Simple sigmoid-like activation function suitable for
digital hardware implementation, \textit{Stat. Methods Med. Res.,} \textbf{28} (1992), 1379--1380, doi:10.1049/el:19920877.
\bibitem{17} J. Han, C. Moraga, The influence of the sigmoid function
parameters on the speed of backpropagation learning, \textit{ Int. Workshop Artif.
Neural Netw.,} \textbf{930} (2005), 195--201.
\bibitem{18} J. Han, R. S. Wilson, S. E. Leurgans, Sigmoidal mixed models for
longitudinal data, \textit{Stat. Methods Med. Res.,} \textbf {27} (2018), 863--875.
https://doi.org/10.1177/0962280216645632.
\bibitem{12} J. Y. Kang, Some Properties and Distribution of the Zeros of the
$q$--Sigmoid Polynomials, \textit{Discret. Dyn. Nat. Soc.,} \textbf{2020} (2020), 1--10.
\bibitem{10} J. Y. Kang, Determination of the properties of $(p,q)$--sigmoid
polynomials and the structure of their roots, In Number Theory and Its
Applications; IntechOpen: London, UK, (2020), 1-19.
http://dx.doi.org/10.5772/intechopen.91862.
\bibitem{kang} J. Y. Kang, Explicit properties of $q$--sigmoid polynomials
combining $q$--cosine function, \textit{J. Appl. Math. Inform.,} \textbf {39} (2021),
541--552.
\bibitem{11} J. Y. Kang, Some relationships between sigmoid polynomials and
other polynomials, \textit{J. Pure Appl. Math.,} \textbf{1} (2019), 57--67.
\bibitem{WC25} W. A. Khan, C. K\i z\i late\c{s}, Some properties of
Frobenius--Sigmoid polynomials, Springer Volume, (accepted).
\bibitem{PAS} O. K. Pashaev, S. Nalci, Golden quantum oscillator and
Binet--Fibonacci calculus, \textit{J. Phys A: Math. Theor.,} \textbf{45} (2011), 1--19..
\bibitem{PAS1} O. K. Pashaev, Quantum calculus of Fibonacci divisors and
infinite hierarchy of bosonic--fermionic golden quantum oscillators, \textit{Internat. J. Geom. Methods Modern Phys.,} \textbf{18} (2021), 2150075.
\bibitem{KROT} E. Krot, An introduction to finite fibonomial calculus,
\textit{Centr. Eur. J. Math.,} \textbf{2} (2004), 754-766.
\bibitem{OZVATAN} M. \"{O}zvatan, Generalized golden--Fibonacci calculus and
applications, Ph.D. thesis, Izmir Institute of Technology, 2018.
\bibitem{KUS} S. Ku\c{s}, N. Tuglu, T. Kim, Bernoulli $F$--polynomials and
Fibo--Bernoulli matrices, \textit{Adv. Differ. Equ.,} \textbf{2019} (2019), 145.
\bibitem{MKIM} M. S. Kim, Some properties of Fibonacci--sigmoid numbers and
polynomials matrix, \textit{J. Anal. Appl.,} \textbf{21} (2023), 89-99.
\bibitem{MAR24} M. S. Alatawi, W. A. Khan, C. K\i z\i late\c{s}, C. S. Ryoo, Some
Properties of Generalized Apostol--Type Frobenius--Euler--Fibonacci
Polynomials, \textit{Mathematics,} \textbf{12} (2024), 800.
\bibitem{GUAN24} H. Guan, W. A. Khan, C. K\i z\i late\c{s}, C. S. Ryoo, On Certain
Properties of Parametric Kinds of Apostol--Type Frobenius--Euler--Fibonacci
Polynomials, \textit{Axioms}, \textbf{13} (2024), 348.
\bibitem{UR24} A. Urieles, R. William, H. L. C. Perez, M. J. Ortega,
J. Arenas-Penaloza, On $F$--Frobenius--Euler polynomials and their matrix
approach, \textit{J. Math. Comput. Sci.,} \textbf{32} (2024), 377--386.
\bibitem{CAKIR25} B. C. \c{C}ak\i r, E. Erkus-Duman, Appell-Type Changhee
Polynomials in the Framework of Fibonomial Calculus, \textit{Dolomites Res. Notes
Approx.,} \textbf{18} (2025), 56--63.
\bibitem{CKHO23} C. K\i z\i late\c{s}, H. \"{O}zt\"{u}rk, On parametric types
of Apostol Bernoulli--Fibonacci Apostol Euler--Fibonacci and Apostol
Genocchi--Fibonacci polynomials via Golden calculus, \textit{AIMS Math.,} \textbf{8}
(2023), 8386--8402.
\bibitem{KUS19} S. Kus, N. Tuglu, T. Kim, Bernoulli $F$--polynomials and
Fibo--Bernoulli matrices, \textit{Adv. Differ. Equ.,} \textbf{2019} (2019), 1--16.
\bibitem{TUG21} N. Tu\u{g}lu, E. Ercan, Some properties of Apostol Bernoulli
Fibonacci and Apostol Euler Fibonacci Polynomials, Icmee-2021, 32--34.
\bibitem{ORHAN25} O. Di\c{s}kaya, Fibonacci--based generalizations of
degenerate Stirling numbers, \textit{Ramanujan J.,} \textbf{68} (2025), 51.
\bibitem{TUG14} N. Tu\u{g}lu, F. Yesil, E. Gokcen Kocer, M. Dziemia\'{n}czuk, The $%
F$--Analogue of Riordan Representation of Pascal Matrices via Fibonomial
Coefficients, \textit {J. Appl. Math.,} \textbf{2014} (2014), 1--6.
\bibitem {H1} H. Qawaqneh, Fractional analytic solutions and fixed point results with some applications, Adv. Fixed Point Theory, 14(1), 2024.
\bibitem {H2} H. Qawaqneh, M.S.Md. Noorani, H. Aydi, A. Zraiqat, A.H. Ansari, On fixed point results in partial b-metric spaces, Journal of Function Spaces, 2021, 8769190, 9 pages, 2021.
\bibitem {H3} H. Qawaqneh, M.S.Md. Noorani, H. Aydi, Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications, AIMS Mathematics, 8(3),2023, 6682-6696.
\bibitem {H4} H. Qawaqneh, H.A. Hammad and H. Hassen Aydi, Exploring new geometric contraction mappings and their applications in fractional metric spaces, AIMS Mathematics, {\bf9}(1) (2024), 521--541.
\bibitem {H5} H. Qawaqneh, New Functions For Fixed Point Results In Metric Spaces With Some Applications, Indian Journal of Mathematics, {\bf66}(1) (2024), 55--84.
\bibitem{Kanan1} M. Elbes, T. Kanan, M. Alia and M. Ziad, COVD-19 detection platform from X-ray images using deep learning, International Journal of Advances in Soft Computing and its Applications, 14 (1), pp. 197-211, 2022.
\bibitem{Kanan2} T. Kanan, M. Elbes, K. Abu Maria, M. Alia, Exploring the potential of IoT-based learning environments in education, International Journal of Advances in Soft Computing and its Applications, 15 (2), 2023.
\bibitem{Baitha} I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh and Sh. Momani, Design fractional-order PID controllers for single-joint robot arm model, International Journal of Advances in Soft Computing and its Applications, 14(2), 2022, pp. 96-114.
Published
2026-04-18
Section
Special Issue: Advances in Nonlinear Analysis and Applications
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