Algebraic, Differential and Fractional Properties of Balancing-like Polynomials
FRACTAL AND FRACTIONAL, cilt.10, sa.10, ss.732, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 10 Sayı: 10
- Basım Tarihi: 2026
- Doi Numarası: 10.3390/fractalfract10100732
- Dergi Adı: FRACTAL AND FRACTIONAL
- Derginin Tarandığı İndeksler: Academic Search Ultimate (EBSCO), Scopus, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest), Science Citation Index Expanded (SCI-EXPANDED), INSPEC, Directory of Open Access Journals
- Sayfa Sayıları: ss.732
- Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
- Marmara Üniversitesi Adresli: Evet
Özet
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.