Cauchy identity for Jack functions

Question

There are two versions of Cauchy identity for Schur functions, namely $$ \sum_{\lambda}s_\lambda(\underline x)s_\lambda(\underline y) = \prod_{i,j=1}^n\frac 1{1-x_iy_j}\ ,\qquad {\rm (1)} $$ and $$ \sum_{\lambda}s_\lambda(\underline x)s_{\lambda'}(\underline y) = \prod_{i,j=1}^n (1+x_iy_j)\ .\qquad {\rm (2)} $$

(Usual notations employed here: $\underline x=(x_1,\dots,x_n)$, $\lambda$ runs over all partitions (of length $\leq n$), $\lambda'$ is the conjugate partition.)

I found in the literature (Stanley's 1989 paper, Macdonald's book) the following generalization of (1) to Jack symmetric poylnomials: $$ \sum_{\lambda} \frac{J^{(a)}_ \lambda(\underline x)J^{(a)} _ \lambda(\underline y) }{\langle J^{(a)}_ \lambda,J^{(a)}_ \lambda\rangle_ a}= \prod_{i,j=1}^n\biggl(\frac 1{1-x_iy_j}\biggr)^{1/a}\ . $$

(Here, $J^{(a)}_ \lambda$ are the Jack polynomials (in the $J$-normalization) and $\langle,\rangle_a$ the deformed Hall inner product, namely $\langle p_\lambda,p_\mu\rangle_a=a^{\ell(\lambda)}z_\lambda \delta_{\lambda\mu}$.)

QUESTION: does there exist a similar generalization of (2) for Jack polynomials?

Answer 1

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.)