Using $q$-calculus to study $LD{L}^T$ factorization of a certain Vandermonde matrix

We use tools from $q$-calculus to study $LD{L}^T$ decomposition of the Vandermonde matrix $V_q$ with coefficients $v_{i,j}=q^{ij}$. We prove that the matrix $L$ is given as a product of diagonal matrices and the lower triangular Toeplitz matrix $T_q$ with coefficients $t_{i,j}=1\mathrm{/}(q;q{)}_{i-j}$, where $(z;q{)}_k$ is the q-Pochhammer symbol. We investigate some properties of the matrix $T_q$, in particular, we compute explicitly the inverse of this matrix.