A class of non-symmetric band determinants with the Gaussian q-binomial coefficients

Abstract

A class of symmetric band matrices of bandwidth 2r+1 with the binomial coefficients entries was studied earlier. We consider a class of non-symmetric band matrices with the Gaussian q-binomial coefficients whose upper bandwith is s and lower bandwith is r. We give explicit formulæ for the determinant, the inverse (along with its infinity-norm when q →1) and the LU-decomposition of the class. We use the celebrated q-Zeilberger algorithm and unimodality property to prove claimed results.

Keywords: Determinant, inverse matrix, LU-decomposition, Gaussian q-binomial coefficients, Fibonomial coefficients, Zeilberger's algorithm, unimodality