Rigorous inhomogeneous spin Whittaker scaling and orthogonality

Prove rigorously that the inhomogeneous spin q-Whittaker polynomials converge under the specified q\to1 scaling to the inhomogeneous spin Whittaker functions, and establish the corresponding continuous orthogonality identity with the Sklyanin-type density, including the required convergence and decay estimates.

Background

The paper formally introduces inhomogeneous spin Whittaker functions as q\to1 scaling limits of the spin q-Whittaker polynomials. The proposed scaling sends x_i=q^{X_i}, a_i=q^{A_i}, b_i=q^{B_i}, and indexes partitions through \lambda_i=\lfloor\log_q(1/L_i)\rfloor.

The resulting continuous orthogonality relation is stated with a Sklyanin-type density and a delta distribution in the Weyl-chamber variables. The discussion is explicitly non-rigorous because proving the limit requires control of q-Pochhammer asymptotics, weak convergence, and decay of the integrands. Thus both the scaling-limit assertion and the continuous orthogonality formula remain conjectural in the inhomogeneous setting.