  We calculate the $(\mathfrak{g},K)$ module structure for the principal series
representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a
hypergeometric generating function together with an inverse Mellin transform
technique as an improvement to the method to calculate the intertwining
operators. We have shown that the matrix entries of the simple intertwining
operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the
matrix entries of the long intertwining operator can be expressed as the
constant term of the Laurent expansion of some hypergeometric function.
