On the transformation formula of the slice Bergman kernels in the quaternion variables (Q2828679)

Let \(\mathbb{H}\) denote the algebra of real quaternions such that \(q\in\mathbb{H}\) has the form \(q= q_0+ q_1e_1+ q_2e_2+ q_3 e_3\), where \(e_1\), \(e_2\), and \(e_3\) satisfyNEWLINENEWLINE i) \(e_ie_j+ e_je_i=-2\delta_{ij}\) for \(1\leq i\), \(j\leq 3\) andNEWLINENEWLINE ii) \(e_1e_2= e_3\), \(e_2e_3= e_1\), \(e_3e_1= e_2\).NEWLINENEWLINE Let \(\mathbb{S}^2= \{q_1e_1+ q_2e_2+ q_3e_3\in \mathbb{R}^3\mid q^2_1+ q^2_2+ q^2_3= 1\}\). For \(I\) in \(\mathbb{S}^2\), define \(L_I= \{x+ yI\mid x,y\in\mathbb{R}\}\). If \(\Omega^1\) and \(\Omega^2\) are axially symmetric slice domains in \(\mathbb{H}\) and there exists a slice biregular function \(\phi\) on \(\Omega^1\) such that \(\phi(\Omega^1\cap L_I)\subset L_I\) for all \(I\) in \(\mathbb{S}^2\) and \(\phi(\Omega^1)= \Omega^2\), then \textit{F. Columbo} et al. [Complex Var. Elliptic Equ. 57, No. 7--8, 825--839 (2012; Zbl 1251.30055)] derived a transformation formula expressing the Bergman kernel function for the domain \(\Omega^1\cap L_I\) in terms of the Bergman kernel function for the domain \(\Omega^2\cap L_I\).NEWLINENEWLINE In the paper under review the author provides a transformation formula of the Bergman kernel functions for the whole domains \(\Omega^1\) and \(\Omega^2\).