On combinatorial generalization(s) of Borel transform: Averaging method in combinatorics of symmetric polynomials

We elaborate on the recent suggestion to consider averaging of Cauchy identities for Schur functions over power sum variables. This procedure has apparent parallels with Borel transform, only it changes number combinatorial factors like dR in sums Young diagrams instead just factorials ordinary numbers. It provides a universal view previously known, but seemingly random identities.