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Asymptotic behavior of zeros of Bessel function derivatives

Árpád Baricz, Pranav Kumar, Saminathan Ponnusamy

TL;DR

This work derives two complementary asymptotic descriptions for the zeros $j_{ u,k}^{(n)}$ of the $n$-th derivative of Bessel functions. It first establishes a McMahon-type expansion for large $k$ with fixed $ u$, including explicit error bounds, by adapting Olver’s turning-point inversion method to $J_ u^{(n)}$ and its even/odd derivatives. It then develops a uniform large-$ u$ expansion, using Olver’s Airy-function framework to relate $J_ u^{(n)}( u x)$ to Airy functions and to locate zeros via Airy zeros $a_k$, $a'_k$, yielding precise asymptotics like $j_{ u,k}^{(3)}= u- rac{a_k}{2^{1/3}} u^{1/3}+ ceil ext{(higher-order terms)}$ and $j_{ u,k}^{(4)}= u- rac{a'_k}{2^{1/3}} u^{1/3}+ ceil ext{(higher-order terms)}$. The results extend Wong–Lang–Olver analyses and provide a general framework for zeros of derivatives of special functions, with potential extensions to modified Bessel derivatives and related bounds.

Abstract

We derive two distinct asymptotic expansions for the zeros $j_{ν,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_ν^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $ν$, for which we also establish an explicit error bound. The second addresses the case when $ν\to \infty$ with fixed $k$ and it involves the zeros of Airy functions and their derivatives. These results extend and refine the classical work of Wong, Lang, and Olver on the zeros of Bessel functions. In the course of obtaining our main results, we also generalize several auxiliary results, which in turn provide a broader framework for the study of zeros of special functions.

Asymptotic behavior of zeros of Bessel function derivatives

TL;DR

This work derives two complementary asymptotic descriptions for the zeros of the -th derivative of Bessel functions. It first establishes a McMahon-type expansion for large with fixed , including explicit error bounds, by adapting Olver’s turning-point inversion method to and its even/odd derivatives. It then develops a uniform large- expansion, using Olver’s Airy-function framework to relate to Airy functions and to locate zeros via Airy zeros , , yielding precise asymptotics like and . The results extend Wong–Lang–Olver analyses and provide a general framework for zeros of derivatives of special functions, with potential extensions to modified Bessel derivatives and related bounds.

Abstract

We derive two distinct asymptotic expansions for the zeros of the -th derivative of Bessel function . The first is a McMahon-type expansion for the case when with fixed , for which we also establish an explicit error bound. The second addresses the case when with fixed and it involves the zeros of Airy functions and their derivatives. These results extend and refine the classical work of Wong, Lang, and Olver on the zeros of Bessel functions. In the course of obtaining our main results, we also generalize several auxiliary results, which in turn provide a broader framework for the study of zeros of special functions.
Paper Structure (12 sections, 4 theorems, 220 equations)

This paper contains 12 sections, 4 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. McMahon-type expansion for zeros of Bessel functions derivatives
  3. Some initial results
  4. McMahon-type expansion for $j_{\nu, k}^{(n)}$
  5. An error bound
  6. Asymptotic expansions for large $\nu$
  7. Some older results
  8. Expression for $J_\nu^{(n)}(x)$ and $J_\nu^{(n)}(x\nu)$
  9. Asymptotic behaviour of zeros of $J_\nu^{(n)}(x) \text{ and } J_\nu^{(n)}(x\nu)$
  10. Uniform asymptotic expansion for $J_\nu^{(n)}(\nu x)$
  11. Asymptotic expansion of zeros of $J_\nu^{(n)}(\nu x)$
  12. Discussion and future work

Key Result

Theorem 2.1

For $n\in\mathbb{N}_0$ and large $x$, the $2n$-th derivative of the Bessel function of first kind $J_\nu(x)$, can be expressed as where and Moreover, the $(2n+1)$-th derivative of $J_\nu(x)$ can be expressed as where and In particular for $n=0$, we obtain that with

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.1
  • proof : Proof of Theorem \ref{['Theorem1']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['Theorem2']}
  • Remark 2.2
  • Lemma 2.1
  • Theorem 3.1
  • Remark 3.1