[Paper Review] On a new class of 2-orthogonal polynomials, II_The integral representations

Khalfa Douak, P. Maroni|arXiv (Cornell University)|Dec 22, 2022

Mathematical functions and polynomials4 citations

TL;DR

This paper establishes integral representations for a new class of 2-orthogonal polynomials by solving a matrix differential equation involving two linear functionals. It derives weight functions supported on the real line or positive real line, expressed via special functions such as confluent hypergeometric and Laguerre functions, with Dirac masses added when necessary to satisfy boundary conditions, thereby completing the characterization of the functionals via explicit measures.

ABSTRACT

A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties. Integral Transforms Spec Funct. 2021;32(2):134-153]. Six interesting special cases were pointed out there. For each case, we precisely deal with the integral representation problem for the functionals associated to these polynomials. The focus is on the matrix differential equation $\big({\bfΦU}\big)'+{\bfΨU}=0$, with ${\bf U}={^t}(u_0 , u_1)$ and ${\bfΦ}$, ${\bfΨ}$ are $2 imes2$ polynomial matrices, from which we establish the differential equations satisfied by the two functionals. Based on this, depending on the case, we show that $u_0$ and $u_1$ are represented via weight functions supported on the real line or positive real line and defined in terms of various special functions. In order for certain integral representations to exist, addition of Dirac mass is necessary.

Motivation & Objective

• To determine integral representations for the linear functionals $u_0$ and $u_1$ associated with a new class of 2-orthogonal polynomials.
• To solve the matrix differential equation $({\bf \Phi U})' + {\bf \Psi U} = 0$ for $2\times2$ polynomial matrices ${\bf \Phi}, {\bf \Psi}$ and vector ${\bf U} = (u_0, u_1)^t$.
• To identify weight functions $w_0$ and $w_1$ supported on the real line or positive real line that satisfy the orthogonality conditions.
• To incorporate Dirac masses when necessary to meet boundary conditions for the differential systems.
• To validate the solutions using normalization and boundary condition constraints, ensuring consistency with the recurrence structure of the polynomials.

Proposed method

• Solve the matrix differential equation $({\bf \Phi U})' + {\bf \Psi U} = 0$ to derive the differential systems satisfied by $u_0$ and $u_1$.
• Reduce the system to second-order linear differential equations for the weight functions $w_0$ and $w_1$, such as $w_0'' - 2s w_0' + w_0 = 0$ or $x w_0' + (x - \alpha) w_0 = 0$.
• Use special functions—such as confluent hypergeometric functions and Laguerre weights—as solutions to the differential equations under specific parameter constraints.
• Apply boundary conditions derived from duality brackets, including $[x w_0 f]_{\mathscr{C}} = 0$ and $[(w_1 - \nu w_0) f]_{\mathscr{C}} = 0$, to constrain the solutions.
• Introduce Dirac masses at $x=0$ when the weight functions fail to satisfy boundary conditions on unbounded domains.
• Normalize the solutions using gamma functions and hypergeometric identities to ensure consistency with the dual sequence and recurrence coefficients.

Experimental results

Research questions

• RQ1 Can the linear functionals $u_0$ and $u_1$ associated with the 2-orthogonal polynomials be represented via integrals with respect to weight functions on the real line or positive real line?
• RQ2 What differential equations do the weight functions $w_0$ and $w_1$ satisfy, and how do their solutions depend on the parameters $r$, $s$, $\alpha_1$, and $\beta_0$?
• RQ3 Under what conditions do the boundary conditions for the functionals hold, and when is the addition of Dirac masses necessary for valid integral representations?
• RQ4 How do the solutions involving confluent hypergeometric and Laguerre functions arise from the differential systems?
• RQ5 Can the normalization constants and dual sequence conditions be consistently satisfied using the derived weight functions?

Key findings

• For the case $2 + p + q = 0$, the weight functions are $w_0(x) = c e^{\delta_1 x}$ for $x \geq 0$ and $w_0(x) = c e^{\delta_0 x}$ for $x \leq 0$, with $c = -1/(2r)$, and $w_1(x)$ defined piecewise using exponential and polynomial terms.
• In Subcase VI.2 with $\eta = 0$, $\delta_1 = -1$, $\beta_0 = \alpha + 1$, and $\nu = \delta_0^{-1}$, the weight $w_0(x) = \frac{1}{\Gamma(\alpha+1)} x^\alpha e^{-x}$ is the classical Laguerre weight function.
• The weight $w_1(x)$ in Subcase VI.2 is given by $w_1(x) = \nu w_0(x) - \frac{\nu^2 e^{\nu x}}{(\nu+1)^{\alpha+1}} \frac{\gamma(\alpha+1, (\nu+1)x)}{\Gamma(\alpha+1)}$, valid for $\alpha \geq 0$ and $-1 < \nu < 0$, with the lower incomplete gamma function.
• Boundary conditions $[x w_0 f]_{\mathscr{C}} = 0$ and $[(w_1 - \nu w_0) f]_{\mathscr{C}} = 0$ are satisfied under the stated parameter constraints, ensuring consistency with the duality pairing.
• Normalization constants are explicitly computed as $\lambda_0 = -k / (\beta_0 \Gamma(-p))$ and $\lambda_1 = \delta_1 k / (2r(1+p)\Gamma(-p))$, with $k$ involving the hypergeometric function ${}_2F_1$.
• The solution for the case $p = q = -1$ yields $w_0(x) = c_1 e^{\delta_1 x} + c_2 e^{\delta_0 x}$ on $\mathbb{R}$, with $c_1, c_2$ arbitrary, and $w_1$ determined via $\eta w_1 = -x w_0' + \delta_0(x - \beta_0 - \delta_1) w_0$.