On the explicit Binet formula of the generalized ${{2}^{nd}}$ orders Recursive relation

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Journal of Universal Mathematics
Cilt: 8 Sayı: 2
On the explicit Binet formula of the generalized ${{2}^{nd}}$ orders Recursive relation

On the explicit Binet formula of the generalized ${{2}^{nd}}$ orders Recursive relation

Pages : 133-140

View : 37 | Download : 28

Publication Date : 2025-10-19

Article Type : Research Paper

Abstract :In this paper, second-order generalized linear recurrence relations of the form ${{V}_{n}}\\\\left( {p}_{1},{p}_{2}, {V}_{1},{V}_{2}\\\\right)={{p}_{1}}{{V}_{n-1}}+{p}_{2}{{V}_{n-2}}$ , where ${{p}_{1}},{{p}_{2}},$ ${{V}_{1}}\\\\left( =a \\\\right)$ and $ {{V}_{2}}\\\\left( =b \\\\right) $ are arbitrary integers, are studied to derive Binet-like formulas in simplified and comprehensive generalized forms. By imposing specific constraints on the coefficients $\\\\left( {{p}_{1}},{{p}_{2}} \\\\right)$ and the initial terms $\\\\left( {{V}_{1}},{{V}_{2}} \\\\right)$, various well-known existing formulas, such as those for classical Fibonacci and Lucas sequences, emerge as special cases of this generalization.
Keywords : ${{2}^{nd}}$ Sıra Özyinelemeli İlişkiler, Genelleştirilmiş Üreten Fonksiyonlar, Açık Binet Formülleri

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