Staircase hook-length ratios and special values of Jacobi polynomials

We relate hook-length products for adjacent staircase partitions to special values of Jacobi polynomials. This connection expresses the number of semistandard tableaux in terms of Jacobi polynomials defined via Gauss hypergeometric functions. From this identity, we derive the special values of stable Grothendieck polynomials and $K$-theoretic Schur $P$-functions indexed by adjacent staircase partitions. These values provide ratios of the numbers of set-valued and shifted set-valued semistandard tableaux. This connection is further clarified by the theory of excited Young diagrams, which characterizes the coefficients in these specializations.