In this paper, we consider three systems of coupled generalized Sylvester quaternion equations$$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1} \\ A_{2}X_{2}-Y_{1}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}, \right. \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2}\\ A_{3}Y_{2}-Y_{3}B_{3}=C_{3} \end{array}, \right. \end{aligned}$$and$$\begin{aligned} \left\{ \begin{array}{c} A_{1}X_{1}-Y_{1}B_{1}=C_{1}\\ A_{2}Y_{1}-Y_{2}B_{2}=C_{2} \\ A_{3}X_{2}-Y_{2}B_{3}=C_{3} \end{array}. \right. \end{aligned}$$We present new necessary and sufficient conditions for the solvability of each system, and derive the determinantal representations of the general solutions to the above systems by the row and column determinants of quaternion matrices.

中文翻译：

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广义Sylvester方程组解的行列式表示。

在本文中，我们考虑了三个耦合的广义Sylvester四元数方程组$$ \ begin {aligned} \ left \ {\ begin {array} {c} A_ {1} X_ {1} -Y_ {1} B_ {1} = C_ {1} \\ A_ {2} X_ {2} -Y_ {1} B_ {2} = C_ {2} \\ A_ {3} X_ {2} -Y_ {2} B_ {3} = C_ {3} \ end {array}，\ right。\ left \ {\ begin {array} {c} A_ {1} X_ {1} -Y_ {1} B_ {1} = C_ {1} \\ A_ {2} Y_ {1} -Y_ {2} B_ {2} = C_ {2} \\ A_ {3} Y_ {2} -Y_ {3} B_ {3} = C_ {3} \ end {array}，\ right。\ end {aligned} $$和$$ \ begin {aligned} \ left \ {\ begin {array} {c} A_ {1} X_ {1} -Y_ {1} B_ {1} = C_ {1} \ \ A_ {2} Y_ {1} -Y_ {2} B_ {2} = C_ {2} \\ A_ {3} X_ {2} -Y_ {2} B_ {3} = C_ {3} \ end {数组}。\对。\ end {aligned} $$我们为每个系统的可解性提供了新的必要和充分的条件，并通过四元数矩阵的行和列行列式得出了上述系统的一般解的行列式表示。