We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski.
The question here is: is there any other variant of Schur Polynomial which also has similar properties as Schur and Shifted Schur functions?
Edits:
What I mean by similar properties are things like:
a) this function satisfies Vanishing Theorem
b) let $s_{\mu}^* (\lambda)$ be a shifted Schur polynomials, then these polynomials can be written in some forms like: $f(\lambda) = s^*_ \mu (\lambda) \times g(n)$ where $g(n)$ is a function of $n$.
c) this function has some recursion equations
Sorry if the question is still too general enough...