  One of spectacular results in mathematical physics is the expression of Racah
matrices for symmetric representations of the quantum group $SU_q(2)$ through
the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions
${_4\phi_3}$. Recently it was shown that this is in fact the general property
of symmetric representations, valid for arbitrary $SU_q(N)$, at least for
exclusive Racah matrices $\bar S$. The natural question then is what
substitutes the conventional $q$-hypergeometric polynomials when
representations are more general? New advances in the theory of matrices $\bar
S$, provided by the study of differential expansions of knot polynomials,
suggest that these are multiple sums over Young sub-diagrams of the one, which
describes the original representation of $SU_q(N)$. A less trivial fact is that
the entries of the sum are not just the factorized combinations of quantum
dimensions, as in the ordinary hypergeometric series, but involve
non-factorized quantities, like the skew characters and their further
generalizations -- as well as associated additional summations with the
Littlewood-Richardson weights.
