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 $R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }$, $R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }$, $n=0,1,2,...$. By the result of Hendriksen and Van Rossum there exists a linear functional $\mathbf{L}$ on Laurent polynomials, such that $\mathbf{L}(R_n R_m) = 0$, when $n\not= m$, while $\mathbf{L}(R_n^2)\not= 0$. 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.