Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

We investigate the asymptotic behavior of the Selberg-like integral $$\frac{1}{N!}{\int }_{[0,1{]}^N}{x}_1^p\prod _{i<j}(x_i-x_j{)}^2\prod _i{x}_i^{a-1}(1-x_i{)}^{b-1}d{x}_i\text{,}$$ as $N\to \mathrm{\infty }$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton’s interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.