  In a $m$ particle quantum system, one can have $k=1,\,2,\,\ldots,\,m$ body
interactions. The rank of interactions and the nature of particles (fermions or
bosons) can strongly affect the dynamics of the system. To explore this in
detail, we study quenched quantum dynamics in many particle systems varying
rank of interactions, both for fermionic and bosonic particles. We represent
the system Hamiltonian by Fermionic Embedded Gaussian Orthogonal Ensembles
(FEGOE) and Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE)
respectively. We show that generating function for $q$-Hermite polynomials
describes the semi-circle to Gaussian transition in spectral densities of
FEGOE$(k)$ and BEGOE$(k)$ (also the Unitary variants FEGUE$(k)$ and BEGUE$(k)$)
as a function of rank of interactions $k$. Importantly, numerical Fourier
transform of generating function of $q$-Hermite polynomials explains the
short-time decay of survival probability in FEGOE$(1+k)$ and BEGOE$(1+k)$. The
parameter $q$ describing these properties is related to excess parameter
$\gamma_2$ and we give a formula for FEGOE$(k)$, FEGUE$(k)$, BEGOE$(k)$ and
BEGUE$(k)$. We illustrate that the dynamics strongly depends on the rank of
interactions and nature of particles and these universal features may be
relevant to modeling non-equilibrium quantum systems.
