Algebraic, Differential and Fractional Properties of Balancing-like Polynomials
- 1Department of Mathematics, Faculty of Science, Marmara University, Istanbul 34722, Türkiye
Abstract
This study investigates the algebraic and differential properties of balancing-like polynomials, which constitute a polynomial extension of balancing-like sequences. Several fundamental properties are established, including matrix representations, addition and subtraction formulas, and polynomial analogues of the Cassini, Catalan, D’Ocagne, and Vajda identities. A product factorization is obtained, which yields an explicit description of the zeros of these polynomials. Moreover, a direct connection with the Chebyshev polynomials of the second kind is established, and the ordinary generating function of the balancing-like polynomials is derived. The derivative sequences associated with these polynomials are then systematically investigated. Explicit formulas, recurrence relations, and a self-convolution identity are obtained, together with a relation recovering the original polynomials from their neighboring derivative terms. Finally, the analysis is extended to higher-order derivatives, and a generating-function representation and a general recurrence relation for the resulting differential hierarchy are derived. The differential framework is further extended to non-integer orders by means of the Caputo fractional derivative. An explicit fractional derivative formula and a generating-function representation are obtained.
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BibTeX
@article{ozkan2026algebraic,
title = {{Algebraic, Differential and Fractional Properties of Balancing-like Polynomials}},
author = {Engin Özkan and Beyza Tekeoğlu Akkaya},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10100732},
url = {https://www.mdpi.com/2504-3110/10/10/732},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Algebraic, Differential and Fractional Properties of Balancing-like Polynomials
AU - Engin Özkan
AU - Beyza Tekeoğlu Akkaya
PY - 2026
DA - 2026/10/21
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100732
UR - https://www.mdpi.com/2504-3110/10/10/732
AB - This study investigates the algebraic and differential properties of balancing-like polynomials, which constitute a polynomial extension of balancing-like sequences. Several fundamental properties are established, including matrix representations, addition and subtraction formulas, and polynomial analogues of the Cassini, Catalan, D’Ocagne, and Vajda identities. A product factorization is obtained, which yields an explicit description of the zeros of these polynomials. Moreover, a direct connection with the Chebyshev polynomials of the second kind is established, and the ordinary generating function of the balancing-like polynomials is derived. The derivative sequences associated with these polynomials are then systematically investigated. Explicit formulas, recurrence relations, and a self-convolution identity are obtained, together with a relation recovering the original polynomials from their neighboring derivative terms. Finally, the analysis is extended to higher-order derivatives, and a generating-function representation and a general recurrence relation for the resulting differential hierarchy are derived. The differential framework is further extended to non-integer orders by means of the Caputo fractional derivative. An explicit fractional derivative formula and a generating-function representation are obtained.
SN - 2504-3110
ER -