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...

Enumerative Combinatorics, vol. 2. – Richard Stanley Mar 27 '13 at 1:30