On the symmetry properties of the PT kernels and recurrence relations

Perturbation Theory to Large Scale Structure Cosmology proposes corrections to the linearly evolved density contrast and velocity in terms of a series development in which all terms are integrals of powers of the linear density contrast multiplied by kernels. We discuss the symmetry properties of these kernels and show that their full symmetrized versions can be decomposed in different classes of subkernels. We will construct classes of subkernels with improved symmetry properties, and provide recurrence relations to generate them.