Fourier coefficients of non-analytic automorphic functions of several variables (Q1069974)

This paper generalizes a classical formula for the mean \(\sum_{n\leq x}a_ n (x-n)^{\rho}\) of the Fourier coefficients of a holomorphic cusp form [cf. \textit{K. Chandrasekharan} and \textit{R. Narasimhan}, Ann. Math., II. Ser. 74, 1-23 (1961; Zbl 0107.037)]. The automorphic forms considered in the present paper live on hyperbolic \((k+1)\)-space and are invariant under groups of transformations as considered by \textit{H. Maass} [Abh. Math. Semin. Univ. Hamb. 16, No.3/4, 72-100 (1949; Zbl 0034.343)]. The Fourier coefficients are parametrized by lattice points in this case. The quantity studied is \[ \sum_{0<| \beta |^ 2\leq x}a_{\beta} P(\beta) (x-| \beta |^ 2 )^{\rho}, \] with P a harmonic polynomial. It is expressed as a series over all in the lattice, the terms contain \(a_{\beta}\) and complicated special functions.