## 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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### References:

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