(Asked in MSE but got no response)

The generating function $\frac{1}{(1-t)^N}=\sum_k {N+k-1\choose k}t^k=\sum_k h_k(1)t^k$ and the Jacobi-Trudi formula $s_{\lambda/\mu}=\det(h_{\lambda_i-i-\mu_j+j})$ tell me that the value of the skew Schur function at the identity is $$ s_{\lambda/\mu}(1_N)=\det\left({N+\lambda_i-i-\mu_j+j-1\choose \lambda_i-i-\mu_j+j}\right).\qquad (1)$$

However, I was reading a paper by Chen and Stanley (A Formula for the Specialization of Skew Schur Functions) and they state that $$s_{\lambda/\mu}(1,q,q^2,...)=\frac{1}{\prod_{u\in\lambda/\mu}[N+c(u)]_q}\det\left(\left\lbrack \begin{matrix} N+\lambda_i-i\\\lambda_i-i-\mu_j+j\end{matrix}\right\rbrack_q\right),\qquad (2)$$ where $c(u)$ is the content of the box $u$ in the Young diagram of shape $\lambda/\mu$ and the $q$-quantities are $[x]_q=1-q^x$, $[a]_q!=[a]_q[a-1]_q\cdots$ and $\left\lbrack \begin{matrix} a\\b\end{matrix}\right\rbrack_q=\frac{[a]_q!}{[b]_q![a-b]_q!}.$

I am not an expert in this $q$-business, and I am confused by this equation. I have a few closely related questions.

1. since the left hand side of (2) is a polynomial in $q$, it should have a limit when $q\to 1$ and this should be the skew Schur at the identity. Is this correct? But what is the number of arguments?

2. The determinant at the right hand side on (2) has a limit when $q\to 1$, but the prefactor does not. How to take the limit $q\to 1$ of this equation?

3. How to obtain equation (1) from equation (2)?

(Asked in MSE but got no response.)

The generating function $\frac{1}{(1-t)^N}=\sum_k {N+k-1\choose k}t^k=\sum_k h_k(1)t^k$ and the Jacobi–Trudi formula $s_{\lambda/\mu}=\det(h_{\lambda_i-i-\mu_j+j})$ tell me that the value of the skew Schur function at the identity is $$ s_{\lambda/\mu}(1_N)=\det\left({N+\lambda_i-i-\mu_j+j-1\choose \lambda_i-i-\mu_j+j}\right).\tag{1}\label{1}$$

However, I was reading a paper by Chen and Stanley (A Formula for the Specialization of Skew Schur Functions) and they state that $$s_{\lambda/\mu}(1,q,q^2,\dotsc)=\frac{1}{\prod_{u\in\lambda/\mu}[N+c(u)]_q}\det\left(\genfrac[]{0pt}{}{N+\lambda_i-i}{\lambda_i-i-\mu_j+j}_q\right),\tag{2}\label{2}$$ where $c(u)$ is the content of the box $u$ in the Young diagram of shape $\lambda/\mu$ and the $q$-quantities are $[x]_q=1-q^x$, $[a]_q!=[a]_q[a-1]_q\cdots$ and $\genfrac[]{0pt}{}a b_q=\frac{[a]_q!}{[b]_q![a-b]_q!}$.

I am not an expert in this $q$-business, and I am confused by this equation. I have a few closely related questions.

1. Since the left hand side of \eqref{2} is a polynomial in $q$, it should have a limit when $q\to 1$ and this should be the skew Schur at the identity. Is this correct? But what is the number of arguments?

2. The determinant at the right hand side on \eqref{2} has a limit when $q\to 1$, but the prefactor does not. How to take the limit $q\to 1$ of this equation?

3. How to obtain equation \eqref{1} from equation \eqref{2}?