Non-linear additive twist of Fourier coefficients of $GL(3) \times GL(2)$ and $GL(3)$ Maass forms

Let λ π ( r , n ) be the Fourier coefficients of a Hecke-Maass cusp form π for SL ( 3 , Z ) and λ 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 λ π ( m , n ) and λ f ( n ) . More precisely, we have

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