Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula $$s_{\lambda} (1,q,\dots,q^{n1}) = \prod_{(i,j) \in \lambda} \frac{1q^{n+ji}}{1q^{\lambda_i + \lambda_j'ij+1}}.$$

There is a very nice specialization when you consider infinitely many variables, due to an identity of Kawanaka: $$Q_{\lambda}(1,q,q^2,\dots)=\prod_{i=1}^m \frac{(1;q)_{\lambda_i}}{(q;q)_{\lambda_i}} \prod_{1\le i < j \le m} \frac{q^{\lambda_j}q^{\lambda_i}}{1q^{\lambda_i+\lambda_j}}$$ where $\lambda$ is a partition of length $m$. For the case of finitely many variables the story is a little more complicated. In the ordinary Schur polynomial case we have an expression which is a ratio of two determinants and is singular at $(1,1,\dots,1)$, however a Vandermonde factorization helps us get rid of the singularity and obtain an expression in $q$, which is also meaningful when $q=1$. The same thing can be done with SchurQ functions. The specialization can be written as a multiple hypergeometric sum, which is singular at $q=1$, but we can find a transformation formula into a hypergeometric sum of Schlosser type which extends to $q=1$. It would be a pain to write out the identities themselves, but if you follow Rosengren's paper on the subject, "Schur Qpolynomials, multiple hypergeometric series and enumeration of marked shifted tableaux", you should have what you want. Combinatorially this is interesting because it gives the $q$enumeration of marked shifted tableaux. 

