ON THE ASYMPTOTICS OF HIGHER POWER MOMENTS OF

HECKE EIGENVALUES OVER TWO SPARSE SEQUENCES OF

POSITIVE INTEGERS

Let f and g be two distinct normalized primitive holomorphic cusp forms of even

integral weights κ1 and κ2 for the full modular group Γ = SL(2, Z), and let λf (n)

and λg(n) denote the n-th normalized Fourier coefficients of f and g, respectively.

In this paper, we establish the asymptotic formulae for the summatory functions

associated to λ2ℓ1

f (n)λ2ℓ2

g (n), with ℓ1, ℓ2 ≥ 2 being any fixed integers, over two

sparse sequences of positive integers, which generalizes the existing results in the

literature in this direction.