A novel beta matrix function via Wiman matrix function and their applications (Q6064176)

A square matrix \(U \in \mathbb{C}^{m\times m}\) is said to be positive stable if the real part of all eigenvalues is positive. For a positive stable matrix \(U\), the gamma matrix function is \[ \Gamma(U) = \int_{0}^{\infty} e^{-t} t^{U-1} dt, \] where \(t^{U-1} := \exp((U-I)\log t)\) and \(I\) denotes the identity matrix of order \(m\). Under certain conditions, also \(\Gamma^{-1}(U)\) is a well-defined matrix, as well as the Pochhammer symbol \((U)_k\) with \(k\) a nonnegative integer, the beta function in matrix form \[ \mathcal{B}(U,V) = \int_0^1 t^{U-I} (1-t)^{V-I} dt, \] the hypergeometric matrix function, the confluent hypergeometric matrix function, and some other matrix functions. The authors study a new beta matrix function that is denoted as \(\mathcal{B}^{(Z);\delta,\zeta}_{(\eta,\xi)}(U,V)\). More precisely, let \(U\), \(V\) and \(Z\) be positive stable and commuting matrices, and assume that the four parameters \(\delta,\zeta,\eta,\xi\) satisfy \(\operatorname{Re}(\eta), \operatorname{Re}(\xi)>0\) and \(\delta,\zeta>0\). Then they define the matrix function \[ \mathcal{B}^{(Z);\delta,\zeta}_{(\eta,\xi)}(U,V) = \int_0^1 t^{U-I} (1-t)^{V-I} E_{(\eta,\xi)}\left( \frac{-Z}{t^\delta(1-t)^\zeta} \right) dt, \] where \(E_{(\eta,\xi)}(\,\cdot\,)\) is the two-parameter Mittag-Leffler matrix function defined by \textit{R. Garrappa} and \textit{M. Popolizio} [J. Sci. Comput. 77, No. 1, 129--153 (2018; Zbl 1406.65031)]. The above function is a valid extension of the hypergeometric and confluent hypergeometric matrix functions.