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Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula $$ H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) . $$ Each $H_k(x)$ is a polynomial of exact ...

There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and $$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for $n\geq1$}.$$ What I have found is the ...

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R. Stanley remarked following ...

Let $\lambda\vdash n$ denote an integer partition of $n$ and $\frak{H}_{\lambda}$ be the multiset of hook lengths of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even ...