Intertwining Operator for $Sp(4,\mathbb{R})$ and Orthogonal Polynomials

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.