Bounding the Gap Between Zeros of the Variable- Parameter Confluent Hypergeometric Function

TL;DR

This paper establishes a lower bound on the spacing between zeros of the confluent hypergeometric function with a variable parameter, and uses it to evaluate asymptotic approximation accuracy for Wiener process first passage probabilities.

Contribution

It introduces a new lower bound on zero spacing for the confluent hypergeometric function with variable parameters, enhancing understanding of its zero distribution.

Findings

Derived a lower bound on zero spacing for $\

Proved monotonicity of the zero spacing bound.

Applied the bound to evaluate asymptotic approximations for Wiener process first passage probability.

Abstract

This paper derives a lower bound on the spacing between adjacent zeros of the confluent hypergeometric function $\Phi(a,b,z)$ when $a$ is variable and $(b,z) \in \mathbb{R}^+$ are known and fixed. Monotonicity of the bound is established, and the results are used to assess the accuracy of asymptotic approximations for the first passage probability of a Wiener process.