Multipoint Hermite-Padé approximations for beta functions (Q1957036)

The authors construct the multipoint Hermite-Padé approximant \(P_n/Q_n\) to the function \(f_\alpha(z)=B(\alpha,z)\) meromorphic on the entire complex plane and having simple poles at the points \(-m\) (\(m\in\mathbb{Z}\)), where \(B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}\) is the beta function. \(P_n/Q_n\) is defined as follows: \(R_n(z)=Q_n(z)f_\alpha(z)-P_n(z)\) and satisfy the interpolation condition \(R_n(\beta +k)=0\) for some \(\beta>0\) and \(k=0, \ldots,2n\). They prove that the uniform limite distribution of the zeros of the polynomials \(Q_n\) exists.