Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\lambda$.
I want to know if following results appear in literature if anyone can give me a reference or proof. $$ \sum_{\text{different hook of size d}} \frac{1}{|\lambda|!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-c(\Box)h} (*) $$ Now summing up all different kind of hook by choosing parameter $\ell$ that is the length of eastward wing of the hook $\lambda$. Also replacing the $dim \lambda$ in terms of $d,\ell$ and the content $c(\Box)$ $$\sum_{\ell=1}^d (-1)^{d-\ell}\frac{1}{d} \binom{d-1}{\ell-1} \prod_{i=1}^d \frac{1}{1-(\ell-i)h}(**)$$ Simfilifying it reduces to \begin{equation} \frac{(2d-2)!}{ d!} \prod_{i=1}^{d-1} \frac{h^{d-1}}{(1+ih)(1-ih)}(***) \end{equation} Now if I replace in (*) $\prod_{\Box \in \lambda} \frac{1}{1-c(\Box)h}$ by $\prod_{\Box \in \lambda}(1+c(\Box)h)^2$ is there any nice form like equaton (***)? In general, if I replace eq (*) with a symmetric function involving content of the hook and $h$ is there any form overall any combinatorial explanation?