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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| Published in: | Mathematics (Basel) Vol. 13; no. 14; p. 2321 |
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| Abstract | 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. |
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| AbstractList | 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. 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 A[sub.n] [sup.2]−A[sub.n+1]A[sub.n−1]=k[sup.−2n], A[sub.n] [sup.2]−A[sub.n]A[sub.n−1]+1/k2A[sub.n−1] [sup.2]=k[sup.−2n], and B[sub.n] [sup.2]−B[sub.n]B[sub.n−1]+1/k2B[sub.n−1] [sup.2]=−(k[sup.2]−4)k[sup.−2n], and some generalizations such as B[sub.n+m] [sup.2]−(k[sup.2]−4)A[sub.n−t]B[sub.n+m]A[sub.t+m]−(k[sup.2]−4)k[sup.2t−2n]A[sub.t+m] [sup.2]=k[sup.−2m−2t]B[sub.n−t] [sup.2], A[sub.t+m] [sup.2]−B[sub.t−n]A[sub.n+m]A[sub.t+m]+k[sup.2n−2t]A[sub.n+m] [sup.2]=k[sup.−2n−2m]A[sub.t−n] [sup.2], and more were derived, where m,n,t∈ℤ and t≠n. In addition to this, the solution pairs of the algebraic equations x[sup.2]−B[sub.p]xy+k[sup.−2p]y[sup.2]=k[sup.−2q]A[sub.p] [sup.2], x[sup.2]−(k[sup.2]−4)A[sub.p]xy−(k[sup.2]−4)k[sup.−2p]y[sup.2]=k[sup.−2q]B[sub.p] [sup.2], and x[sup.2]−B[sub.p]xy+k[sup.−2p]y[sup.2]=−(k[sup.2]−4)k[sup.−2q]A[sub.p] [sup.2] are presented, where A[sub.p] and B[sub.p] are k-Oresme and k-Oresme–Lucas numbers, respectively. 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+ 1k2 An−12=k−2n , and Bn2−BnBn−1+ 1k2 Bn−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. |
| Audience | Academic |
| Author | Demirtürk, Bahar |
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| Cites_doi | 10.1007/s10665-024-10350-6 10.46793/KgJMat2405.747SL 10.1080/00150517.1975.12430693 10.1080/00150517.1990.12429516 10.1080/00029890.1961.11989696 10.1109/IV64223.2024.00014 10.1007/978-0-306-48517-6_10 10.3390/sym16111407 10.1002/9781118033067 10.1016/j.chaos.2006.09.022 10.3934/mbe.2023092 10.20944/preprints202402.0910.v1 10.1080/00150517.2017.12427733 10.1080/00150517.1965.12431416 10.3390/math9070789 10.1080/00150517.2019.12427646 10.1080/00207390903236426 10.9734/ajarr/2023/v17i10532 10.1080/00150517.1997.12428995 10.1007/978-1-4757-4330-2 10.3390/sym17050697 10.1007/b98892 10.3390/math12081156 10.3390/axioms14010014 |
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| RelatedPersons | Oresme, Nicole |
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| SubjectTerms | 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 |
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