I am interested in evaluating the following quantity: $$ _0F_1(b, cI_p)$$
From the definition of the hypergeometric function of the matrix argument, we have that $$ _0F_1(b, X) = \sum_{k = 0}^\infty \sum_{\kappa} \frac{1}{(b)_{\kappa}} C_{\kappa}(X) $$
where $\kappa$ is a partition of the indexes $k$, $C_\kappa(X)$ is called zonal polynomial and $(b)_\kappa$ is the generalized Pochhammer symbol.
Is there any simplification when $X = cI_p$ where $c$ is some constant and $I_p$ is a $p\times p$ identity matrix?