Formulas are developed for evaluating the three-electron integrals, which appear in the transcorrelated method using a frozen Gaussian geminal. First, we derive expressions with intermediates, in which the angular momentum indices of the inter-electronic operators are temporarily increased. Following the inspection on the factorization property, we further derive more straightforward recurrence relations corresponding to the Obara-Saika one for electron repulsion integrals. It is shown that the relations can be applied outside the contraction loops over geminal primitives by introducing three-indexed auxiliary integrals.