@article{article_1595221, title={On the explicit Binet formula of the generalized ${2}^{nd }$ orders Recursive relation}, journal={Journal of Universal Mathematics}, volume={8}, pages={133–140}, year={2025}, DOI={10.33773/jum.1595221}, author={Verma, K. L.}, keywords={${2}^{nd }$ order Recursive relations, generaized generating function, Explicit Binet formulas}, 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.}, number={2}, publisher={Gökhan ÇUVALCIOĞLU}