On Functions of the Second Kind in Orthogonal Polynomial Theory

Abstract

A given weight function \(p(x)\) on an interval \([a, b]\) defines uniquely, subject to normalization, a sequence of orthogonal polynomials \(P_n(x)\) and their corresponding sequence of functions of the second kind $$\begin{aligned} Q_n(x)=\int _a^b\frac{P_n(t)p(t)dt}{x-t},\quad \ x\in \mathbf{C } \backslash [a,b]. \end{aligned}$$ corresponding to them. This paper focuses on the reverse problem of characterizing orthogonal polynomials by means of functions of the second kind, together with the properties of such functions of the second kind. Functions of the second kind for orthogonal polynomials are also of particular interest in that they differ only slightly from the second solution of the differential equation satisfied by the orthogonal polynomials.