On some combinatorial identities related to super Catalan matrix

In this paper, we define a new matrix Sn constructed by super Catalan numbers. Also, we give Cholesky and LU-decompositions, Hermite normal form, and the determinant of the matrix Sn. Moreover, we derive auxiliary results involving some summation formulas via the coefficients of Lucas polynomials and scaled coefficients of Chebyshev polynomials. Additionally, we give a matrix S n by modifying the matrix Sn to deduce a matrix identity related to matrices Sn and Sn. By using the decomposition method, we give an application of solving a system of linear equations of order n with coefficients S (m, n) and find a general solution.