The dual Cauchy identity for Jack polynomials also exists, and is better expressed in terms of the $P$-normalized Jack polynomials: $$ \sum_{\lambda} P^{(a)}_ \lambda(\underline x)P^{(1/a)} _ {\lambda'}(\underline y) = \prod_{i,j=1}^n\bigl(1+x_iy_j\bigr)^{1/a}\ . $$

(Cf., e.g., formula (2.6) in these notes by I.G. Macdonald.)

The dual Cauchy identity for Jack polynomials also exists, and is better expressed in terms of the $P$-normalized Jack polynomials: $$ \sum_{\lambda} P^{(a)}_ \lambda(\underline x)P^{(1/a)} _ {\lambda'}(\underline y) = \prod_{i,j=1}^n\bigl(1+x_iy_j\bigr)\ . $$

(Cf., e.g., formula (2.6) in these notes by I.G. Macdonald.)