Nonlinear Additive Twists of GL(3) x GL(2) and GL(3) Maass Forms

Let lambda(pi) (r, n) be the Fourier coefficients of a Hecke-Maass cusp form pi for SL (3, Z) and lambda(f) (n) be the Fourier coefficients of a Hecke-eigen form f for SL (2, Z). The aim of this article is to get a non-trivial bound on the sum which is non-linear additive twist of the coefficients (lambda(pi) (m, n) and lambda(f) (n). More precisely, we have Sigma(infinity)(n=1) lambda(pi) (r, n) e (alpha n(beta)) V (n/X) <<(pi,epsilon) alpha root beta r(7/6) X3/4+9 beta/28+epsilon. and Sigma(infinity)(n=1)lambda(pi) (r, n) lambda (f) (n) e (alpha n(beta)) V (n/X) <<(pi, f, epsilon) (alpha beta)(3/2) r X3/4 + 29 beta/44 + epsilon, where V (x) is a smooth function supported in [1, 2] and satisfying V-(j) (x) <<(j) 1.