All Questions

Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...

Consider the Fibonacci polynomials $f_n(x)$, defined by the recursion $f_n(x)=xf_{n-1}(x)-f_{n-2}(x)$ with initial values $f_0(x)=0$ and $f_1(x)=1$. It is well known that the values of these ...

Usually, I like working with determinants related to the Vandermonde matrix, i.e. $$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$ However, I run into some unusual matrix and its determinant. Define the $(...

Recalling the $q$-binomials (Gaussian polynomials). Let $[n]_q!=\prod_{j=1}^n\frac{1-q^j}{1-q}$ and $\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}$. Now, consider the $n\times n$ matrix $\mathbf{M}...

The ubiquitous Vandermonde matrix, of entries $(x_i^{j-1})_{i,j}^{1,n}$, and its determinant $$\prod_{i<j}^{1,n}(x_j-x_i)$$ have found many utilities in Combinatorics and Physics, among other ...