Connection coefficients and zeros of orthogonal polynomials

We discuss an old theorem of Obrechkoff and some of its applications. Some curious historical facts around this theorem are presented. We make an attempt to look at some known results on connection coefficients, zeros and Wronskians of orthogonal polynomials from the perspective of Obrechkoffʹs theorem. Necessary conditions for the positivity of the connection coefficients of two families of orthogonal polynomials are provided. Inequalities between the kth zero of an orthogonal polynomial pn(x) and the largest (smallest) zero of another orthogonal polynomial qn(x) are given in terms of the signs of the connection coefficients of the families {pn(x)} and {qn(x)}. An inequality between the largest zeros of the Jacobi polynomials Pn(a,b)(x) and Pn(α,β)(x) is also established.