arXiv subjects

Jorge A. Borrego-Morell

Publications and source records attributed to Jorge A. Borrego-Morell.

4 recordsLinked to original sources

arXiv2026

Semiclassical expansion for exactly solvable differential operators

Below we study a linear differential equation $\MM (v(z,η))=η^M{v(z,η)}$, where $η>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}ρ_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $ρ_M(z)$ is a monic complex-valued polynomial with $\dgr{ρ_M }=M$ and other $ρ_k(z)$'s are complex-valued polynomials with $\dgr{ρ_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $ρ_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense.

math.CA↗

arXiv2026

Liouvillian and Analytic Integrability of a Generalized Gause System

In this work, we identify the regions of the parameter space in which a predator-prey system, derived from the classical Gause model with a generalized Holling response function and logistic prey growth in the absence of predators, fails to be Liouvillian integrable. Although the model parameters have biological meaning only when restricted to appropriate real domains, our analysis is carried out in the complex setting, which provides a unified algebraic framework; the resulting nonintegrability conditions remain valid in the biologically relevant regime. As a consequence, we establish the nonintegrability of an Abel differential equation of the second kind with polynomial coefficients obtained from the system. Finally, we analyze the existence of a local analytic first integral in neighborhoods of the equilibrium points.

math.DS↗

arXiv2024

An asymptotic expansion of eigenpolynomials for a class of linear differential operators

Consider an $M$-th order linear differential operator, $M\geq 2$, $$ \mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $ρ_M $ is a monic complex polynomial such that $degree[ρ_M]=M$ and $(ρ_k)_{k=0}^{M-1}$ are complex polynomials such that $degree[ ρ_k ]\leq k, 0\leq k \leq M-1$. It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure $μ$. We obtain an asymptotic expansion of the eigenpolynomials of $\mathcal{L}^{(M)}$ in compact subsets out the support of $μ$. In particular, we solve a conjecture posed in G.~Masson and B.~Shapiro, ``On polynomial eigenfunctions of a hypergeometric type operator,'' Exper. Math., vol.~10, pp.~609--618, 2001.

math.CA↗

arXiv2022

On orthogonal polynomials with respect to a class of differential operators

We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $\{ρ_k\}_{k=0}^{M}$ are complex polynomials such that $deg[ρ_k]\leq k, 0\leq k \leq M$, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomenon occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials, in terms of a linear system of difference equations with varying coefficients. Also, for the case of a first-order differential operator, we locate the zeros and establish the strong asymptotic behavior of these polynomials.

math.CA↗