We consider the standard hypergeometric differential operator $\mathcal D$ regarded as an operator on the complex plane $\mathbb C$ and the complex conjugate operator $\bar{\mathcal D}$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler–Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.

中文翻译：

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复平面和双谱上的一对交换超几何算子

我们考虑将标准超几何微分算子 $\mathcal D$ 视为复平面 $\mathbb C$ 上的算子和复共轭算子 $\bar{\mathcal D}$。这些算子正式通勤，并根据适当的权重正式地相互邻接。我们找到它们在纳尔逊意义上通勤时的条件，并明确地写出它们的联合谱分解。它由 Jacobi 变换的二维对应物决定（同义词：广义 Mehler-Fock 变换、Olevskii 变换）。我们还表明，逆变换是一对根据虚方向位移定义的交换差分算子的谱分解算子。