content_image/320838.jpg content_image/320837.jpg content_image/320836.jpg content_image/320835.jpg
Substituting $_{0}$F ( α ) 0 ( A , Z ) and r ∏ j =1 ∏ p s $_{=1}$(1 − b$_{s}$z$_{j}$ ) − 1 /α in the above formula by their expansions (6) and (36) in the series of Jack polynomials, and interchang- ing the order of integration and summation, which is possible because the series converge uniformly over the unit torus, we obtain 

<img src='content_image/320838.jpg'>

But ∏ r j $_{=1}$( z$_{j}$ ) ( p − r +1) /α − $^{1}$C ( α ) τ ( Z ) = C ( α ) ˜ τ ( Z ) , where ˜ τ denotes partition [ τ$_{1}$ + p − r +1 − α α , ..., τ$_{r}$ + p − r +1 − α α ]. Note that ˜ τ is well defined for α = 2 /β, where β is an even integer. If β is an odd integer, we need to assume that p − r + 1 is even. Therefore, using the orthogonality of the Jack polynomials with respect to the torus scalar product, we have 

<img src='content_image/320837.jpg'>

Using Lemma A1, (37), and equality r ∏ j =1 a − ( p − r +1) /α +1 j C ( α ) ˜ τ ( A ) = C ( α ) τ ( A ) = C ( α ) τ ( A ) , we get after some cancellations 

where 

<img src='content_image/320836.jpg'>

<img src='content_image/320835.jpg'>

In the above expression for ˜ γ ( α $^{)}$, substitute C ( α ) τ ( I$_{p}$ ) and C ( α ) ˜ τ ( I$_{r}$ ) by their 

