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?



Cauchy identity for Jack functions
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](https://kurims.kyoto-u.ac.jp/EMIS/journals/SLC/opapers/s28macdonald.pdf).)


