We report a comprehensive specific heat and inelastic neutron scattering study to explore the possible origin of multiferroicity in HoCrO$_3$. We have performed specific heat measurements in the temperature range 100 mK - 290 K and inelastic neutron scattering measurements were performed in the temperature range 1.5 - 200 K. From the specific heat data we determined hyperfine splitting at 22.5(2) $mu$eV and crystal field transitions at 1.379(5) meV, 10.37(4) meV, 15.49(9) meV and 23.44(9) meV, indicating the existence of strong hyperfine and crystal field interactions in HoCrO$_3$. Further, an effective hyperfine field is determined to be 600(3) T. The quasielastic scattering observed in the inelastic scattering data and a large linear term $gamma=6.3(8)$ mJmol$^{-1}$K$^{-2}$ in the specific heat is attributed to the presence of short range exchange interactions, which is understood to be contributing to the observed ferroelectricity. Further the nuclear and magnetic entropies were computed to be, $sim$$17.2$ Jmol$^{-1}$K$^{-1}$ and $sim$34 Jmol$^{-1}$K$^{-1}$, respectively. The entropy values are in excellent agreement with the limiting theoretical values. An anomaly is observed in peak position of the temperature dependent crystal field spectra around 60 K, at the same temperature an anomaly in the pyroelectric current is reported. From this we could elucidate a direct correlation between the crystal electric field excitations of Ho$^{3+}$ and ferroelectricity in HoCrO$_3$. Our present study along with recent reports confirm that HoCrO$_3$, and $R$CrO$_3$ ($R=$ Rare earth) in general, possess more than one driving force for the ferroelectricity and multiferroicity.
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