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We present the results of a new study of the K-band galaxy luminosityfunction (KLF) at redshifts z<3.75, based on a nested combination of theUltraVISTA, CANDELS and HUDF surveys. The large dynamic range in luminosityspanned by this new dataset (3-4 dex over the full redshift range) issufficient to clearly demonstrate for the first time that the faint-end slopeof the KLF at z>0.25 is relatively steep (-1.3<alpha<-1.5 for a singleSchechter function), in good agreement with recent theoretical andphenomenological models. Moreover, based on our new dataset we find that adouble Schechter function provides a significantly improved description of theKLF at z<2. At redshifts z>0.25 the evolution of the KLF is remarkably smooth,with little or no evolution evident at faint (M_K>-20.5) or bright magnitudes(M_K<-24.5). Instead, the KLF is seen to evolve rapidly at intermediatemagnitudes, with the number density of galaxies at M_K~-23 dropping by a factorof ~5 over the redshift interval 0.25<z<3.75. Motivated by this, we explore asimple description of the evolving KLF based on a double Schechter functionwith fixed faint-end slopes (alpha_1=-0.5, alpha_2=-1.5) and a sharedcharacteristic magnitude (M_K*). According to this parameterisation, thenormalisation of the component which dominates the faint-end of the KLF remainsapproximately constant, with phi*_2 decreasing by only a factor of ~2 betweenz~0 and z~3.25. In contrast, the component which dominates the bright end ofthe KLF at low redshifts evolves dramatically, becoming essentially negligibleby z~3. Finally, we note that within this parameterisation, the observedevolution of M_K* between z~0 and z~3.25 is entirely consistent with M_K*corresponding to a constant stellar mass of M*~5x10^10 Msun at all redshifts.

Characterising the evolving $K$-band luminosity function using the UltraVISTA, CANDELS and HUDF surveys