The generalized \(k\)-Fibonacci and \(k\)-Lucas numbers. (Q2918605)

The authors of this paper study the following generalization of Fibonacci and Lucas numbers: For any positive real number \(k\) generalized \(k\)-Fibonacci sequence \(\left \{G_{k,n}\right \}_{n\in \mathbb {N}}\) is defined by the recurrent formula NEWLINE\[NEWLINEG_{k, n+1}=k G_{k,n}+G_{k, n-1},\; n\geq 1NEWLINE\]NEWLINE with initial conditions \(G_{k,0}=a\), \(G_{k,1}=b,\) where \(a,b\in \mathbb {R}\).NEWLINENEWLINEIf \(a=b\), \(b=1\) we have \(k\)-Fibonacci sequence, if \(a=2\), \(b=1\) then we have \(k\)-Lucas sequence.NEWLINENEWLINEThe authors prove five theorems. First theorem: Let \(\alpha _{1}\) and \(\alpha _{2}\) are the roots of the equation \(\alpha ^{2}=k\alpha +1\) and \(\alpha _{1}>\alpha _{2}\). Then for \(X=\frac {a+b\alpha _{1}}{\alpha _{1}}\) and \(Y=\frac {a+b\alpha _{2}}{\alpha _{2}}\) we have the Generalized Binet Formula: NEWLINE\[NEWLINEG_{k,n}=\frac {X\alpha ^{n}_{1}-Y\alpha ^{n}_{2}}{\alpha _{1}-\alpha _{2}}NEWLINE\]NEWLINE and the second theorem: NEWLINE\[NEWLINE\lim _{n\rightarrow \infty }\frac {G_{k,n}}{G_{k,n-1}}=\alpha _{1}.NEWLINE\]