Unit delay operator as parameter in function
With $A(q^{-1},t)$ as given in the formula you get

$$A(1,t)=\prod_{k=1}^n(1-2\cos(k\hat\omega_0(t))+1)\tag{1}$$

and

$$A(\rho,t)=\prod_{k=1}^n(1-2\cos(k\hat\omega_0(t))\rho+\rho^2)\tag{2}$$

So $K(t)$ is simply the quotient of (1) and (2).