Orthogonal polynomials---centroid of their zeroes (Q1014761)

Given and orthogonal polynomial sequence \((P_n)_n\), the centroid of the zeros of the \(n\)-th polynomial \(P_n\) is defined by \(s_n=\frac1n\sum_{k=1}^n x_{n,k}\), where \(x_{n,k}\), \(k=1,2,\dots,n\) are the roots of \(P_n\). In the present paper the author reviews some properties of the zeros and the centroid (such as inequalities, invariance with respect to derivatives and primitives) and present several interesting examples and conjectures. Some discussion on the centroid of the first associated polynomials and some results on the so-called Newton summation formula \(S_p(n)=\sum_{k=1}^n (x_{n,k})^p\) are also presented. Finally, few comments on how to extend the study to \textit{discrete} polynomials are included.