New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements...

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Vydané v:Mathematics (Basel) Ročník 13; číslo 14; s. 2321
Hlavný autor: Demirtürk, Bahar
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Basel MDPI AG 01.07.2025
Predmet:
Algebra
Algorithms
Data compression
Eigenvalues
Eigenvectors
generalized Fibonacci sequences
Identities
k-Oresme sequences
Linear algebra
Mathematicians
Matrix algebra
Matrix representation
Number theory
Numbers
Numerical analysis
Oresme, Nicole
Polynomials
Sequences
solutions of algebraic equations
ISSN:2227-7390, 2227-7390
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Shrnutí:Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities An2−An+1An−1=k−2n, An2−AnAn−1+1k2An−12=k−2n, and Bn2−BnBn−1+1k2Bn−12=−(k2−4)k−2n, and some generalizations such as Bn+m2−(k2−4)An−tBn+mAt+m−(k2−4)k2t−2nAt+m2=k−2m−2tBn−t2, At+m2−Bt−nAn+mAt+m+k2n−2tAn+m2=k−2n−2mAt−n2, and more were derived, where m,n,t∈ℤ and t≠n. In addition to this, the solution pairs of the algebraic equations x2−Bpxy+k−2py2=k−2qAp2, x2−(k2−4)Apxy−(k2−4)k−2py2=k−2qBp2, and x2−Bpxy+k−2py2=−(k2−4)k−2qAp2 are presented, where Ap and Bp are k-Oresme and k-Oresme–Lucas numbers, respectively.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:2227-7390
2227-7390
DOI:10.3390/math13142321