The ring of symmetric functions $\Lambda$ is defined as the ring of formal power series of bounded degree in coefficients $X_1,X_2...$ which are invariant under permutation of the coefficients. Given an integer $n$ and a partition $\lambda$ which partitions $n$, the symmetric function $h_\lambda$ is defined as $h_\lambda=\sum_{\bf{i}}X_{i_1}^{\lambda_1}X_{i_2}^{\lambda_2}...X_{i_k}^{\lambda_k}$, where $k$ is the number of parts in the partition and $\bf{i}$ runs over all $k$-tuples of distinct positive integers.
For example, if $n=2$ and $\lambda$ is the partition $(1,1)$ then $h_\lambda=\sum_{i<j}X_iX_j$.
It is easy to see that the $h_\lambda$ form a basis for the symmetric functions.
We also have an inner product on the symmetric functions defined by $$\big \langle h_{\lambda}, h_{\mu} \big \rangle = \delta_{\lambda,\mu}.$$ Where $\delta_{\lambda,\mu}$ is the Kronecker delta.
I want to know whether we have a similar inner product in the algebra of shifted symmetric functions, i.e. whether $$\big \langle h^*_{\lambda}, h^*_{\mu} \big \rangle = \delta_{\lambda,\mu}.$$
A shifted symmetric function is defined similarly to a symmetric function but, instead of being invariant under permutations of the coefficients, a shifted symmetric function is invariant under permutations of the shifted coefficients $X_1-1+ct,X_2-2+ct,...$.
Let $p_i^*=\sum_{j\ge 1}(X_j-j+\frac{1}2)^k-(\frac{1}2-j)^k$. Then $p_i^*$ is a shifted symmetric function. Given a partition $\lambda$, the shifted symmetric function $h^*_\lambda$ is defined as the product $h^*_\lambda=p^*_{\lambda_1}p^*_{\lambda_2}...p^*_{\lambda_k}$.