Generalized binomial expansion on complex matrix space (Q1269082)

Let \(V=\text{Herm}(r\mathbb{C})\) be the set of all \(r\times r\) complex Hermitian matrices and \(\Omega\subset V\) be the set of all positive-definite matrices. For an arbitrary spherical function \(\varphi_{-\mu}\) on \(\Omega\) with index \(\mu\in\mathbb{C}^r\) the author obtains the generalized binomial expansion of \(\varphi_{-\mu}(I_r-x)\) into a spherical series on the open unit ball \(D\) for the spectral norm in \(\mathbb{C}^{r\times r}\), where \(x\in\Omega\) and \(I_r\in\Omega\) is the identity matrix. As one of the corollaries the spherical transform of \(\varphi_{-\mu}(I_r+x)\) is computed.