On orthogonal Laurent polynomials related to the partial sums of power series

Abstract

Let \(f(z) = \sum_{k=0}^\infty d_k z^k\), \(d_k\in\mathbb{C}\backslash\{ 0 \}\), \(d_0=1\), be a power series with a non-zero radius of convergence \(\rho\): \(0 <\rho \leq +\infty\). Denote by \(f_n(z)\) the \(n\)-th partial sum of \(f\), and \(R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }\), \(R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }\), \(n=0,1,2,\dots\). By the general result of Hendriksen and Van Rossum there exists a unique linear functional \(\mathbf{L}\) on Laurent polynomials, such that \(\mathbf{L}(R_n R_m) = 0\), when \(n\not= m\), while \(\mathbf{L}(R_n^2)\not= 0\), and \(\mathbf{L}(1)=1\). We present an explicit integral representation for \(\mathbf{L}\) in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.