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

TL;DR

This paper explores orthogonal Laurent polynomials related to partial sums of power series, providing explicit integral representations and analyzing finite systems using generating function techniques.

Contribution

It introduces explicit integral representations for the orthogonal Laurent polynomials associated with power series partial sums, extending previous theoretical results.

Findings

Explicit integral representation for the linear functional 

Orthogonality properties of Laurent polynomials derived from power series

Analysis of finite systems of such Laurent polynomials

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.