A note on zeros of derivatives of ultraspherical Bessel functions

TL;DR

This paper studies the positive zeros of derivatives of scaled ultraspherical Bessel functions, providing asymptotic expansions for their zeros as the zero index increases, which enhances understanding of their zero distribution.

Contribution

It derives asymptotic formulas for the positive zeros of derivatives of scaled ultraspherical Bessel functions, a novel contribution to special functions analysis.

Findings

Asymptotic expansions for zeros as k→∞

Insights into zero distribution of derivatives

Extension of classical Bessel function zero analysis

Abstract

For any fixed $\nu\ge 0, \delta\in \mathbb R$ and $x>0$, we investigate the positive zeros of the derivatives $j'_{\nu,\delta}(x)$ and $y'_{\nu,\delta}(x)$, where \begin{equation*} j_{\nu,\delta}(x)=x^{-\delta}J_{\nu}(x)\quad\text{and} \quad y_{\nu,\delta}(x)=x^{-\delta}Y_{\nu}(x). \end{equation*} We derive asymptotic expansions for their $k$-th positive zeros as $k\rightarrow \infty$.