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

Jorge A. Borrego-Morell

Abstract

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.

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

Abstract

Consider an -th order linear differential operator, , where is a monic complex polynomial such that and are complex polynomials such that . It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure . We obtain an asymptotic expansion of the eigenpolynomials of in compact subsets out the support of . 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.
Paper Structure (11 sections, 14 theorems, 162 equations, 1 figure)

This paper contains 11 sections, 14 theorems, 162 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main notation and statement of the results
  3. Overview of the proof and structure of the manuscript
  4. The existence of linearly independent exponential solutions
  5. Asymptotic expansion
  6. Stokes phenomenon
  7. Applications
  8. A conjecture on a class of differential operators
  9. Asymptotic behavior of the monic eigenpolynomials of a fourth order differential operator
  10. Concluding Remarks
  11. Acknowledgments

Key Result

Theorem 1

Let $\mathcal{L}^{(M)}, M\geq 2$ be a non--degenerate exactly solvable operator and $Q_{n}$ be the $n$--th monic eigenpolynomial of $\mathcal{L}^{(M)}$, then as $n\rightarrow\infty$ uniformly in compacts subsets $K\subset \Omega$, here $C_j\in \mathcal{H}(\Omega), j\geq 1$ and the symbol $\sim$ has the same meaning as in Olv74 for an asymptotic expansion in the sense of Poincaré.

Figures (1)

  • Figure 1: Regions $\Delta(z^{\prime},\eta), S_+(z_1), S_-(z_1)$

Theorems & Definitions (18)

  • Theorem 1
  • Remark 1
  • Theorem 2: HSib99, Si58
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • ...and 8 more