We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function
for the Macdonald polynomials, being constructed in the tensor product of the
ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the
degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of
$K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the
tableau sum formula of Macdonald polynomials. Next, we demonstrate that the
integer level representation of the Ding-Iohara quantum algebra naturally
produces the currents of the deformed $\mathcal{W}$ algebra.