We report on near-infrared medium-resolution spectroscopy of a sample of luminous and ultra luminous infrared galaxies (LIRGs-ULIRGs), carried out with SOFI at the ESO 3.5m New Technology Telescope. Because of wavelength dependence of the attenuation, the detection of the Pa_alfa or Br_gamma line in the Ks band should provide relevant constraints on SFR and the contribution of an AGN. We find, however, that the intensities of the Pa_alfa and Br_gamma lines, even corrected for slit losses, are

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... average only 10% and 40%, respectively, of that expected from a normal starburst of similar bolometric luminosity. The corresponding star formation rates, after correcting for the attenuation derived from the NIR-optical emission line ratios, are 14% and 60% of that expected if the far infrared luminosity were entirely powered by the starburst. This confirms the existence of a recombination photon deficit, particularly in the case of the Pa_alfa line, already found in the Br_gamma line in other infrared galaxies of similar luminosity. In discussing the possible causes of the discrepancy, we find unlikely that it is due to the presence of an AGN, though two objects show evidence of broadening of the Pa_alfa line and of the presence of coronal line emission. In fact, from our own observations and data collected from the literature we argue that the studied galaxies appear to be predominantly powered by a nuclear starburst. Two scenarios compatible with the present data are that either there exists a highly attenuated nuclear star forming region, and/or that a significant fraction of the ionizing photons are absorbed by dust within the HII regions. We suggest that observations in the Br_alpha spectral region could constitute a powerful tool to disentangle these two possibilities.