The Fourier expansion of Eisenstein series for 𝐺𝐿(3,𝑍)

Imai, K.; Terras, A.

doi:10.1090/S0002-9947-1982-0667167-5

The Fourier expansion of Eisenstein series for $\textrm {GL}(3, \textbf {Z})$
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by K. Imai and A. Terras

Trans. Amer. Math. Soc. 273 (1982), 679-694

DOI: https://doi.org/10.1090/S0002-9947-1982-0667167-5

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Abstract:

The Fourier expansions of Eisenstein series for ${\text {GL}}(3,{\mathbf {Z}})$ are obtained by two methods—one analogous to the classical method used by many number theorists, including Weber, in his derivation of the Kronecker limit formula. The other method is analogous to that used by Siegel to obtain Fourier expansions of Eisenstein series for the Siegel modular group. The expansions involve matrix argument $K$-Bessel functions recently studied by Tom Bengtson. These $K$-Bessel functions are natural generalizations of the ordinary $K$-Bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates.

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