In the ring of shifted symmetric functions $\Lambda^\*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^\*\_k=\sum\_{i=1} \left((x\_i-i+1/2)^k-(-i+1/2)^k\right)$$
(the factor 1/2 is convenient when the $x\_i$ are the parts of a partition, but can replaced by any constant). For a partition $\kappa$ we let $p^\*\_\kappa=p^\*\_{\kappa\_1}p^\*\_{\kappa\_2}\cdots.$ Another shifted analogue to the power sums are the functions $p\_\kappa^\#$, defined by
$$p^\#\_\kappa = \sum\_{\lambda \vdash k} \chi\_\kappa^\lambda s\_\lambda^\*,$$
where k=$|\kappa|$ is the size of the partition $\kappa$. Both $p\_\kappa^\*$ and $p\_\kappa^\#$ provide a basis for $\Lambda^\*$. (For more details: see the original paper 'Shifted Schur Functions' by Okounkov and Olshanski or the post [On shifted symmetric power sums](https://mathoverflow.net/questions/214551/on-shifted-symmetric-power-sums)).


In the literature there is a vast amount of results relating different bases of $\Lambda^\*$ to each other. However, the only relation between the bases by $p\_\kappa^\*$ and $p\_\kappa^\#$ I can find is that
$$p\_\kappa^\#=p\_\kappa^\*+\textrm{lower degree terms},$$
(because the highest degree terms of $p\_\kappa^\#$ and $p\_\kappa^\*$ equal the ordinary symmetric power sum $p\_\kappa$). How can these lower degree terms be expressed in terms of $p\_\kappa^\*$ (or $p\_\kappa^\#$)?


Question: Are there any expressions known relating $p\_\kappa^\*$ and $p\_\kappa^\#$? 



On two types of shifted symmetric power sums
The answer to this question is precisely given by the Gromov-Witten/Hurwitz correspondence by Okounkov and Pandharipande (see <https://arxiv.org/abs/math/0204305>). Up to the constant $(1-2^{-k})\zeta(-k)$ the functions $p\_k^\*$ equal the renormalized shifted symmetric power sums $\mathbf{p}\_k$ which occur in expressions of integrals of descendants classes in Gromov-Witten theory (equation 0.25). The functions $p\_\kappa^\#$ equal (up to a multiplicative constant depending on $\kappa$) the central characters $\mathbf{f}\_\kappa$ of the symmetric group as explained in the related post [On shifted symmetric power sums](https://mathoverflow.net/questions/214551/on-shifted-symmetric-power-sums?rq=1). These functions occur naturally in the character formula for extended Hurwitz numbers (equation 0.10). The completion coefficients relating $\mathbf{p}\_k$ and $\mathbf{f}\_\kappa$ are given by equation 0.22 in the before-mentioned article. 


