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On the hypergeometric matrix function. (English) Zbl 0933.33004

L. Jódar and J. C. Cortés obtained some properties of beta and gamma functions [Appl. Math. Lett. 11, No. 1, 89-93 (1998)]. In this paper, \(F(A,B,C;z)\) is introduced as a matrix power series and conditions for its convergence on the boundary of the unit disc are discussed. It is shown that if \(P\) and \(Q\) are commuting matrices in \(C^{r\times r}\) such that for all integers \(n\geq 0\), one satisfies the condition that \(P+nI\), \(Q+nI\) and \(P+Q+ nI\) are invertible then \(B(P,Q)= \Gamma(P)\Gamma(Q) \Gamma^{-1}(P+Q)\) and that if \(A\), \(B\), \(C\) are positive stable matrices in \(C^{r\times r}\) such that \(\beta(C)> \alpha(A)+ \alpha(B)\) then the series \(E(A,B,C;z)= \sum{(A)_n (B)_n (C)^{-1}_n\over n!}\) is absolutely convergent for \(|z|= 1\), where \(\alpha(A)= \max\{\text{Re}(z); z\in\sigma(A)\}\) and \(\beta(A)= \min\{\text{Re}(z); z\in\sigma(A)\}\), where \(\sigma(A)\) denotes the set of all eigenvalues of \(A\). It is also shown that if \(B\) and \(C\) commute, then \(F(A,B,C;z)\) is a solution of the differential equation \[ z(1- z)W''- zAW'+ W'(C-z(B+I))- AWB= 0,\quad 0\leq|z|< 1 \] satisfying \(F(A,B,C;0)= 1\). Finally, an integral representation of the hypergeometric function is also given.

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
33C90 Applications of hypergeometric functions
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