In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are orthogonal


$$\langle J^\alpha\_\lambda(z\_1,z\_2),J^\alpha\_\mu(z\_1,z\_2)\rangle'\_2=\frac{1}{2}\int\_T J^\alpha\_\lambda(z\_1,z\_2)\overline{J^\alpha\_\mu(z\_1,z\_2)}\prod\_{i\neq j}\left(1-\frac{z\_i}{z\_j}\right)^{1/\alpha}dz^2$$


where the integration contour is $T=\{(z\_1,z\_2)\in\mathbb{C}^2:|z\_1|=1,|z\_2|=1\}$. Therefore, the integral equals $c\_{\lambda,\alpha}\delta\_{\mu,\lambda}$. However, Macdonald does not give the normalization $c\_{\lambda,\alpha}$ for the scalar product. Is the normalization known?



Normalization of Jack polynomial integral-scalar product?
Yes, my friend. Take $J\_\lambda^{(\alpha)}$ in the J-normalization. Let $n$ be the number of variables (which for you is $2$). Let $\lambda'$ denote the conjugate partition to $\lambda$.
Define
$$
C\_\lambda^{(\alpha)}=\prod\_{(i,j) \in \lambda}(\alpha(\lambda\_i-j)+\lambda\_j'-i+1)(\alpha(\lambda\_i-j)+\lambda\_j'-i+\alpha)
$$
and
$$
\mathcal{N}\_\lambda^{\alpha}(n)=\prod\_{(i,j) \in \lambda} \frac{n+(j-1)\alpha-(i-1)}{n+j\alpha-i}.
$$
In this notation,
$$
\int \cdots \int \frac{d\theta\_1}{2 \pi} \cdots \frac{d\theta\_n}{2 \pi} |J\_\lambda^{(\alpha)}|^2 \prod\_{1 \leq j < k \leq n} |e^{i \theta\_k}-e^{i \theta\_j}|^{2/\alpha}= \mathcal{N}\_\lambda^{\alpha}(n) C\_\lambda^{(\alpha)} \frac{\Gamma(1+n/\alpha)}{\Gamma(1+1/\alpha)^n}. 
$$
[Moments of traces of circular beta-ensembles, Tiefeng Jiang and Sho Matsumoto](https://projecteuclid.org/euclid.aop/1449843631 "Moments of traces of circular beta-ensembles")


