TITLE: On shifted symmetric power sums
QUESTION [12 upvotes]: 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?

REPLY [5 votes]: The super Schur function $s_\lambda(x/y)$ is obtained by applying to
$s_\lambda(x,y)$ (a Schur function in two sets of variables
$x=(x_1,x_2,\dots)$ and $y=(y_1,y_2,\dots)$) the algebra homomorphism
$\omega_y\colon \Lambda(x,y)\to\Lambda(x)\otimes \Lambda(y)$, where
$\Lambda(z)$ is the ring of symmetric functions in the variables $z$,
defined by 
  $$ \omega_yp_n(x,y)=p_n(x)+(-1)^{n-1}p_n(y). $$
Equivalently,
   $$ s_\lambda(x/y) =
   \sum_{\mu\subseteq\lambda}s_\mu(x)s_{\lambda'/\mu'}(y). $$
Thus $f_\lambda(x)=s_\lambda(x_1-1,x_2-2,\dots/1,2,3,\dots)$.