• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.945-954
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• 2015

Let ${\lambda}_f(n)$ denote the n-th normalized Fourier coefficient of a primitive holomorphic form f for the full modular group ${\Gamma}=SL_2({\mathbb{Z}})$. In this paper, we are concerned with ${\Omega}$-result on the summatory function ${\sum}_{n{\leqslant}x}{\lambda}^2_f(n^2)$, and establish the following result ${\sum}_{\leqslant}{\lambda}^2_f(n^2)=c_1x+{\Omega}(x^{\frac{4}{9}})$, where $c_1$ is a suitable constant.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.781-794
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• 2008

Let $\mathbb{H}_g$ and $\mathbb{D}_g$ be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let $\mathbb{C}^{(h,g)}$ be the Euclidean space of all $h{\times}g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ onto the Siegel-Jacobi space $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ which gives a partial bounded realization of $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ by $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. and $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$. are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. invariant under the natural action of the Jacobi group $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ explicitly.