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?



On shifted symmetric power sums
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 $f\_\lambda(x)=s\_\lambda(x\_1-1,x\_2-2,\dots/1,2,3,\dots)$.


