Sums of coefficients of the associated Dirichlet series for triple product <i>L</i>-functions in the level aspect

In this paper, we investigate the structures of certain triple product L-functions in the level aspect, and explicitly compute the exact quotients between them and the associated Dirichlet series, where, for the latter, the fundamental pre-condition of the suitable functional equations as assumed in the previous work of Friedlander–Iwaniec [Summation formulae for coefficients of L-functions, Canad. J. Math. 57 (2005) 494–505] will not be satisfied any more. As a result, we study the average behavior of the coefficients of these Dirichlet series, by establishing the power-saving estimates for the sums of them with the level parameters explicitly determined, and obtain the quantitative estimates for nonlinear exponential sums involving the Rankin–Selberg L-functions corresponding to the configurations [Formula: see text] and [Formula: see text].