Behaviours of non-holomorphic Eisenstein series and the theory of q-hypergeometric functions

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Let $E_k(s;z)$ be the non-holomorphic Eisenstein series with an even weight $k$ attached to the modular group $SL_2({\mathbb{Z})$. One of the major achievements of the present project is to establish its complete asymptotic expansion as $\Im z\to+\infty$ ; this yields various useful results on $E_k(s;z)$, which include its asymptotic expansion as $z\to0$ through the sector $0<\arg z<\pi$, its Fourier series expansion, and further the direct proof that it becomes the eigenfunction of the non-Euclidean Laplacian of weight $k$.