Pieri formulae and specialisation of super Jacobi polynomials

We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra $\mathfrak{osp}(2m+1,2n)$ can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties. The first one is that they are eigenfunctions of CMS operator and the second one is that they satisfy the Pieri formulae. As by product we get some interesting identities involving a Young diagram and rational functions. We hope that our approach can be useful in many similar cases.