Determinants and inverses of skew and skew left circulant matrices involving the \(k\)-Fibonacci numbers in communications. I (Q2844408)

The \(k\)-Fibonacci numbers are defined by the recurrence relation NEWLINE\[NEWLINEF_{k,n+1}=kF_{k,n}+F_{k,n-1},\,\,\, F_{k,0}=0,\,\,\, F_{k,1}=1.NEWLINE\]NEWLINE Several authors obtained explicit formulas for determinants of circulant [\textit{D. V. Jaiswal}, Fibonacci Q. 7, 319--330 (1969; Zbl 0191.04501)] and skew circulant [\textit{S.-Q. Shen, J.-M. Cen} and \textit{Y. Hao}, Appl. Math. Comput. 217, No. 23, 9790--9797 (2011; Zbl 1222.15010)] matrices whose elements are the generalized Fibonacci numbers. The purpose of the present paper is to obtain better results for the determinants and inverses of skew circulant and skew left cirulant matrices using some properties of the \(k\)-Fibonacci numbers.