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Numerical algorithms for the real zeros of hypergeometric functions

Amparo Gil, Wolfram Koepf, Javier Segura

TL;DR

The paper tackles the problem of computing all real zeros of hypergeometric function solutions to second-order ODEs. It develops a global fixed-point iteration framework built from first-order difference-differential equations (DDEs) relating the target function to a contrast function with interlaced zeros, and analyzes multiple DDE families across hypergeometric types to achieve robust convergence. By exploiting oscillatory conditions and selecting DDEs that minimize the nonlocal transformation parameters, the authors derive practical selection criteria that yield fairly uniform convergence for all zeros over parameter ranges. The resulting methodology enables efficient, numerically stable zero-finding for $_0F_1$, confluent hypergeometric, and Gauss hypergeometric functions, with explicit guidance on which DDEs to apply in different x-intervals and parameter regimes. These insights significantly improve the reliability and speed of real-zero computations in applications spanning physics and numerical analysis, and the accompanying Maple tools automate DDE generation and fixed-point implementations.

Abstract

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.

Numerical algorithms for the real zeros of hypergeometric functions

TL;DR

The paper tackles the problem of computing all real zeros of hypergeometric function solutions to second-order ODEs. It develops a global fixed-point iteration framework built from first-order difference-differential equations (DDEs) relating the target function to a contrast function with interlaced zeros, and analyzes multiple DDE families across hypergeometric types to achieve robust convergence. By exploiting oscillatory conditions and selecting DDEs that minimize the nonlocal transformation parameters, the authors derive practical selection criteria that yield fairly uniform convergence for all zeros over parameter ranges. The resulting methodology enables efficient, numerically stable zero-finding for , confluent hypergeometric, and Gauss hypergeometric functions, with explicit guidance on which DDEs to apply in different x-intervals and parameter regimes. These insights significantly improve the reliability and speed of real-zero computations in applications spanning physics and numerical analysis, and the accompanying Maple tools automate DDE generation and fixed-point implementations.

Abstract

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.

Paper Structure

This paper contains 20 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Oscillatory conditions
  4. Hypergeometric functions; selection of the optimal DDEs
  5. Selection of the optimal DDEs
  6. Analysis of Hypergeometric functions
  7. Hypergeometric function $_0\hbox{F}_1 (;c;x)$
  8. First DDE
  9. Second DDE
  10. Comparison between DDEs
  11. Confluent hypergeometric function
  12. $(k,m)=(1,0) \rightarrow y_n=\phi$($\alpha +n$; $\gamma$ ; x)
  13. $(k,m)=(0,-1)\rightarrow y_n=\phi$($\alpha$; $\gamma -n$ ; x)
  14. $(k,m)=(1,1)\rightarrow y_n=\phi$($\alpha +n$; $\gamma +n$ ; x)
  15. Comparing fixed point iterations
  16. ...and 5 more sections

Key Result

Theorem 2.1

If $z_{-1}<z_0<z_1$ are three consecutive zeros of $y_n (x(z))$ and $\eta_i (z)\dot{\tilde{\hbox{A}}}_n (z)>0$ in $(z_{-1}, z_{1})$ then $z_{j}=\lim_{n\rightarrow\infty}T^{(n)}(z_0+\Delta z_0)$ where $\Delta z_0=z_0-z_{-j}$, $j=\hbox{sign}(\eta)$. The convergence is monotonic.

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3