We present a theoretical analysis of the gain characteristics of InGaN/AlGaN quantum dot (QD) lasers. We calculate the elastic strain and piezoelectric filed distribution caused by the lattice mismatch between the QD and the barrier using an original method which takes into account the hexagonal symmetry of the structure's elastic properties. The method is based on an analytical derivation of the Fourier transform of the strain tensor and piezoelectric potential. The proposed approach is combined with a plane-wave expansion method to calculate the spectrum and wave functions of the carriers localized in QDs. The many-body gain of a laser containing a periodic array of QDs is calculated using the Pade approximation. We show that band gap reduction and the Coulomb enhancement of the interband transition probability can significantly modify the gain spectrum in InGaN/AlGaN QD lasers.