C ( α ) κ ( x$_{1}$, ..., x$_{p}$ ) is chosen so that Note that $_{0}$F ( α ) 0 ( A, B ) depends on A and B only through a and b. Therefore, in what follows, without loss of generality, we will consider only diagonal matrices A =diag ( a$_{1}$, ..., a$_{p}$ ) and B =diag ( b$_{1}$, ..., b$_{p}$ ) . We will allow a$_{j}$ and b$_{j}$ to be complex, thus extending definition (6) to complex diagonal matrices A and B . As was mentioned in the introduction, for α = 2 , 1 and 1 / 2 , hypergeometric functions $_{0}$F ( α ) 0 ( A, B ) admit the integral representation where G ( α $^{)}$( p ) is the orthogonal group O ( p ) for α = 2 , the unitary group U ( p ) for α = 1 , and the compact symplectic group S p ( p ) for α = 1 / 2. For real diagonal A and B, such a representation follows from the fact that ∫ G ( α $^{)}$( p $_{)}$C ( α ) κ ( AGBG − $^{1}$) (d G ) = C ( α ) κ ( A ) C ( α ) κ ( B ) C ( α ) κ ( I$_{p}$ ) (see Proposition 5.5 of Gross and Richards, 1987), and the fact that ∑ ∞ k =0 ∑ κ ⊢ k 1 k $_{!}$C ( α ) κ ( AGBG − $^{1}$) = e tr ( AGBG − 1 ) , which follows from (7). For complex diagonal A and B , the representation holds by the analytic continuation because both parts of equality (8) are complex analytic functions of the diagonal elements of A and B . Lemma 1 Let A = diag ( a$_{1}$, ..., a$_{p}$ ) and B = diag ( b$_{1}$, ..., b$_{p}$ ) , where a$_{j}$ and b$_{j}$ are real or complex numbers. Assume that a$_{j}$ = 0 for 1 ≤ j ≤ r and a$_{j}$ = 0 for r < j ≤ p , and denote the upper left block of A, diag ( a$_{1}$, ..., a$_{r}$ ) , as A . Further, let Z =diag ( z$_{1}$, ..., z$_{r}$ ) , where z$_{j}$ are complex variables, and let K be a contour in the complex plane that encircles b$_{1}$, ..., b$_{p}$ counter-clockwise. Finally, let α = 2 /β, The main result of this section is as follows. 10 $_{0}$F ( α ) 0 ( A, B ) = ∫ G ( α $^{)}$( p ) e tr ( AGBG − 1 ) (d G ) , (8) ( x$_{1}$ + ... + x$_{p}$ ) k = ∑ κ ⊢ k C ( α ) κ ( x$_{1}$, ..., x$_{p}$ ) . (7)