Some properties of r-circulant matrices with k-balancing and k-Lucas balancing numbers

Let us define the r-circulant matrix Cr=Circr(c0,c1,c2,…,cn-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ C_r = Circ_r(c_0, c_1,c_2,\ldots ,c_{n-1})$$\end{document} such that the entries of Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_r $$\end{document} are ci=Bk,s+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i=B_{k,s+it}$$\end{document} or ci=Ck,s+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i=C_{k,s+it}$$\end{document}, where Bk,s+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{k,s+it}$$\end{document} and Ck,s+it\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{k,s+it}$$\end{document} are k-balancing and k-Lucas balancing numbers, respectively having arithmetic indices. In this study, we investigate the eigenvalues, determinants, Euclidean norm and bounds estimation for the spectral norm and spread of these matrices and obtain some new identities for the given sequences. Moreover, we extend the study to the corresponding right circulant and skew-right circulant matrices.