The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{k_2}...$.

We known that the sum $\frac{1}{n!}\sum_{\mu\vdash n}C_\mu \chi_\lambda(\mu)p_\mu(x)=s_\lambda(x)$ returns Schur functions (where $C_\mu$ is the size of a conjugacy class in the permutation group and $\chi$ are the irreducible characters of this group).

My question is: what is known about the result of $\frac{1}{n!}\sum_{\mu\vdash n}C_\mu \chi_\lambda(\mu)p_\mu^*(x)$? Is it related to shifted Schur functions? What properties does it share with the usual Schur functions?

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{k_2}...$.

We know that the sum $\frac{1}{n!}\sum_{\mu\vdash n}C_\mu \chi_\lambda(\mu)p_\mu(x)=s_\lambda(x)$ returns Schur functions (where $C_\mu$ is the size of a conjugacy class in the permutation group and $\chi$ are the irreducible characters of this group).

My question is: what is known about $f_\lambda(x):=\frac{1}{n!}\sum_{\mu\vdash n}C_\mu \chi_\lambda(\mu)p_\mu^*(x)$? Is this related to shifted Schur functions? What properties does this function share with the usual Schur functions?