It is shown that the coefficients of the expansion of the parabolic subbases of the two-dimensional hydrogen atom with respect to the polar subbases can be expressed in terms of the generalized hypergeometric function $_3 F_2$ for argument value $x=1$. The expansions of the elliptic basis with respect to the polar and the parabolic are also investigated. A study is made of the limits $R\to 0$ and $R\to\infty$ ($R$ is the parameter which occurs in the definition of the elliptic coordinates) in the expansions of the elliptic basis, and expressions for the coefficients of the expansions of the elliptic basis in terms of the elliptic separation constant are obtained.