  We suggest a further generalization of the hypergeometric-like series due to
M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly
arbitrary function. Moreover, this generalization is valid for the entire
Shiraishi series, not only for its Noumi-Shiraishi part. The theta function
needed in the recently suggested description of the double-elliptic systems, 6d
N=2* SYM instanton calculus and the doubly-compactified network models, is a
very particular member of this huge family. The series depends on two kinds of
variables, $\vec x$ and $\vec y$, and on a set of parameters, which becomes
infinitely large now. Still, one of the parameters, $p$ is distinguished by its
role in the series grading. When $\vec y$ are restricted to a discrete subset
labeled by Young diagrams, the series multiplied by a monomial factor reduces
to a polynomial at any given order in $p$. All this makes the map from
functions to the hypergeometric-like series very promising, and we call it
Shiraishi functor despite it remains to be seen, what are exactly the morphisms
that it preserves. Generalized Noumi-Shiraishi (GNS) symmetric polynomials
inspired by the Shiraishi functor in the leading order in $p$ can be obtained
by a triangular transform from the Schur polynomials and possess an interesting
grading. They provide a family of deformations of Macdonald polynomials, as
rich as the family of Kerov functions, still very different from them, and, in
fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov
case, these polynomials do not depend on the ordering of Young diagrams in the
triangular expansion.
