Summary: We investigate the several special functions defined by a matrix integral on the Hermitian matrix space of size n. They are the matrix argument analogues of the Gauss hypergeometric, Kummer's confluent hypergeometric, the Bessel, the Hermite-Weber and Airy functions which play important roles in the multivariate statistical analysis and the random matrix theory. We give the integral representations for them as functions of eigenvalues of the matrix argument by using the result of Harish-Chandra and Itzykson-Zuber, and give the systems of differential equations for them. We show that these system are holonomic and have the holonomic rank \(2^n\) using the theory of Gröbner basis.