We investigate the large $N$ behavior of the smallest eigenvalue, $\lambda_{ N}$, of an $\left(N+1\right)\times \left(N+1\right)$ Hankel (or moments) matrix $\mathcal{ H}_{ N}$, generated by the weight $w(x)=x^{ \alpha}(1-x)^{ \beta},~x\in[0,1],~ \alpha>-1,~\beta>-1$. By applying the arguments of Szeg\"{ o}, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials $P_{ n}(z),z\in\mathbb{ C}\setminus[0,1]$, associated with $w(x)$, which are required in the determination of $\lambda_{ N}$. Based on this formula, we produce the expressions for $\lambda_{ N}$, for large $N$. Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large $N$.
Categories: math-ph math.MP