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

Steven Langel

Abstract

This paper derives a lower bound on the spacing between adjacent zeros of the confluent hypergeometric function $Φ(a,b,z)$ 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.

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

Abstract

This paper derives a lower bound on the spacing between adjacent zeros of the confluent hypergeometric function when is variable and 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.

Paper Structure

This paper contains 16 sections, 14 theorems, 115 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions and Preliminary Results
  3. Properties of $a^*$
  4. Determining an Analytic Bound
  5. Lower Bound on Zero Separation
  6. Monotonicity of the Bound
  7. Discussion
  8. First Passage Problem
  9. Exact False Alarm Probability in Terms of Residues
  10. Bounding the False Alarm Probability
  11. Numerical Generation of Probability Bounds
  12. Results
  13. Conclusion
  14. Integral Derivations
  15. Inverse Laplace Transform as a Residue Expansion
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\Phi(a, b, z_l)$ be the confluent hypergeometric function of the first kind, where $(b, z_l) \in \mathbb{R}^+$ are known and fixed, and let $a_k^* < a_{k-1}^*$ be two consecutive real zeros of $\Phi(a,b,z_l)$. Then with $g_k = e^{2\pi / \sqrt{(b-2a_k^*)^2 - (b-1)^2}}$ and $\beta_k = b - a_k^* -

Figures (6)

  • Figure 1: Labeling scheme for zero sequences $a^*$ and $z^*$.
  • Figure 2: Qualitative depiction of trajectories followed by the zeros $a^*$ as $z$ varies.
  • Figure 3: Trajectories of two consecutive zeros.
  • Figure 4: Monotonicity threshold value $\bar{a}^*$ for $b \geq 0.32$.
  • Figure 5: Example zero landscape.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 17 more