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

Year 2025, Volume: 8 Issue: 2, 133 - 140, 19.10.2025

K. L. Verma

https://doi.org/10.33773/jum.1595221

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

${{2}^{nd}}$ order Recursive relations , generaized generating function , Explicit Binet formulas

Project Number

None

Burada genelleştirilmiş ${{2}^{nd}}$ emir özyinelemeli ilişkisinin açık Binet formülü verilmiştir.

Abstract

Bu yazıda, ${{V__{n}}\left( {p__{1},{p__{2}, {V__{1} formundaki ikinci dereceden genelleştirilmiş doğrusal yineleme ilişkileri ,{V__{2}\right)={{p__{1}}{{V__{n-1}}+{p__{2}{{V__{n-2} }$ , burada ${{p__{1}},{{p__{2}},$ ${{V__{1}}\left( =a \right)$ ve $ {{V} _{2}}\left( =b \right) $ rastgele tamsayılardır ve basitleştirilmiş ve kapsamlı genelleştirilmiş formlarda Binet benzeri formüller türetmek için incelenmiştir. $\left( {{p__{1}},{{p__{2}} \right)$ katsayılarına ve $\left( {{V__{1}) başlangıç ​​terimlerine belirli kısıtlamalar uygulayarak },{{V__{2}} \right)$, klasik Fibonacci ve Lucas dizileri için olanlar gibi iyi bilinen çeşitli mevcut formüller, bu genellemenin özel durumları olarak ortaya çıkar.

Keywords

${{2}^{nd}}$ Sıra Özyinelemeli İlişkiler , Genelleştirilmiş Üreten Fonksiyonlar , Açık Binet Formülleri

Project Number

None

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