Consider $h_k(x)$ the complete homogeneous symmetric polynomials and $p_k(x)$ the power sum polynomials for some set of variables $x$. If there a finite number of non-zero $p_k$, is there an identity that expresses $\sum_{k=0}^\infty h_k^2$ in terms of $p_k$?
If we take only $p_1 $ different from zero, one gets
$$\sum_{k=0}^\infty h_k^2 = \sum_{j=0}^\infty \frac{(p_1)^{2k} }{ (k!)^2 } = I_0 (2 p_1)~, $$
where $I_0 (2 p_1)$ is the modified Bessel function of the first kind. If we take more $p_k$ to be non-zero, the calculation quickly becomes more complicated, so I am wondering whether a more general expression in terms of $p_k$ is known.