## Hypergeometric functions of several matrix arguments: a preliminary report |

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### Contents

Mathematical Preliminaries | 1 |

4a Some Basic Properties of Appells Functions | 13 |

7a Some Integral Representations | 22 |

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1.1a Section Appell's functions Definition 6.7 denoted differential elements eigenvalues establish the result established by expanding evaluated fA(a,bi Feriet functions following result functions of matrix gamma function gamma integral hence the result Humbert's functions hypergeometric function independent real variables integral representation integral to establish Jacobian matrix Kampe de Feriet Laplace transform large number Lauricella functions limiting forms lower triangular matrix M-transform with respect Mathai and Saxena matrix of functionally Mellin transform negative integer nonsingular matrix Note p x p real symmetric parameters positive definite matrices proof of Theorem Proof Take properties R(bj R(dj real symmetric positive reduction formulae replaced right side rp(a scalar variable Section 1.1 Section skew symmetric Stirling's formula symmetric matrix symmetric positive definite Take the M-transform Theorem Theorem 3.4 Tp(a Tp(c type 1 Dirichlet type 2 beta u}xu wedge product zonal polynomials