Let ${\left\langle\matrix {n \cr k}\right\rangle}_r$ denote the $r-$Hoggatt binomials defined by $${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \langle n-k \rangle_r!}}$$ with ${\langle n\rangle}_r!= {\langle 1\rangle}_r\dots{\langle n\rangle}_r$, where ${\langle n\rangle}_r=\binom{n+r-1}{r}$ and let $D$ be the differentiation operator.
The well-known formula $$\frac{1}{k!}D^k {\frac{1}{1-x}}=\sum_{n\geq0}\binom{n+k}{k}x^n$$ generalizes to a formula for the generating function of the $r-$Hoggatt binomials
$$\prod_{j=0}^{r-1}\frac{j!}{(k+j)!}(x^{r-1}D^r)^k \frac{1}{1-x}=\sum_{n\geq0}{\left\langle\matrix {n+k \cr k}\right\rangle}_r x^{n+r-1}.$$
An explicit expression of the $r-$Hoggatt binomials is $${\left\langle\matrix {n \cr k}\right\rangle}_r= \prod_{1 \le i \le k} \frac{(n - i + 1)^{\overline{r}}}{(k - i + 1)^{\overline{r}}}$$ where $x^{\overline{r}}$ denotes the rising factorial $x(x + 1) \dots (x + r-1).$
The right-hand side can be interpreted as the number of semistandard Young tableaux with shape $r^k.$
Questions: Does the above formula for the generating function of the $r-$Hoggatt binomials occur in the literature about Young tableaux? Has it a combinatorial interpretation or proof?
