  In this study, we introduce a novel analytical Gaussian Process (GP)
cosmography methodology, leveraging the differentiable properties of GPs to
derive key cosmological quantities analytically. Our approach combines cosmic
chronometer (CC) Hubble parameter data with growth rate (f) observations to
constrain the $\Omega_{\rm m0}$ parameter, offering insights into the
underlying dynamics of the Universe. By formulating a consistency relation
independent of specific cosmological models, we analyze under a flat FLRW
metric and first-order Newtonian perturbation theory framework. Our analytical
approach simplifies the process of Gaussian Process regression (GPR), providing
a more efficient means of handling large datasets while offering deeper
interpretability of results. We demonstrate the effectiveness of our
methodology by deriving precise constraints on $\Omega_{\rm m0}h^2$, revealing
$\Omega_{\rm m0}h^2=0.139\pm0.017$. Moreover, leveraging $H_0$ observations, we
further constrain $\Omega_{\rm m0}$, uncovering an inverse correlation between
mean $H_0$ and $\Omega_{\rm m0}$. Our investigation offers a proof of concept
for analytical GP cosmography, highlighting the advantages of analytical
methods in cosmological parameter estimation.
