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Variations on hypergeometric functions

Michał Zakrzewski, Henryk Żołądek

TL;DR

This work develops new integral representations for generalized and confluent hypergeometric functions via multidimensional residues and hypercube integration, enabling robust WKB analyses in multiple regimes. It then analyzes variations of hypergeometric functions under small perturbations and offers a streamlined proof of Wasow’s theorem, clarifying that the Airy equation and its perturbation are analytically equivalent without invoking WKB/Stokes phenomena. A major theme is the connection between hypergeometric equations and generating functions for Multiple Zeta Values, yielding new differential equations, monodromy insights, and corrections to prior claims about $\Delta_3(\lambda)$. The results deepen the understanding of special function theory, mirror-symmetry links, and the differential-analytic structure underlying MZVs.

Abstract

We prove new integral formulas for generalized hypergeometric functions and their confuent variants. We apply them, via stationary phase formula, to study WKB expansions of solutions: for large argument in the confuent case and for large parameter in the general case. We also study variations of hypergeometric functions for small perturbations of hypergeometric equations, i.e., in expansions of solutions in powers of a small parameter. Next, we present a new proof of a theorem due to Wasow about equivalence of the Airy equation with its perturbation; in particular, we explain that this result does not deal with the WKB solutions and the Stokes phenomenon. Finally, we study hypergeometric equations, one of second order and another of third order, which are related with two generating functions for MZVs, one $Δ_2 (λ)$ for $ζ(2, \ldots , 2)$'s and another $Δ_3 (λ)$ for $ζ(3, \ldots , 3)$'s; in particular, we correct a statement from [ZZ3] that the function $Δ_3(λ)$ admits a regular WKB expansion.

Variations on hypergeometric functions

TL;DR

This work develops new integral representations for generalized and confluent hypergeometric functions via multidimensional residues and hypercube integration, enabling robust WKB analyses in multiple regimes. It then analyzes variations of hypergeometric functions under small perturbations and offers a streamlined proof of Wasow’s theorem, clarifying that the Airy equation and its perturbation are analytically equivalent without invoking WKB/Stokes phenomena. A major theme is the connection between hypergeometric equations and generating functions for Multiple Zeta Values, yielding new differential equations, monodromy insights, and corrections to prior claims about . The results deepen the understanding of special function theory, mirror-symmetry links, and the differential-analytic structure underlying MZVs.

Abstract

We prove new integral formulas for generalized hypergeometric functions and their confuent variants. We apply them, via stationary phase formula, to study WKB expansions of solutions: for large argument in the confuent case and for large parameter in the general case. We also study variations of hypergeometric functions for small perturbations of hypergeometric equations, i.e., in expansions of solutions in powers of a small parameter. Next, we present a new proof of a theorem due to Wasow about equivalence of the Airy equation with its perturbation; in particular, we explain that this result does not deal with the WKB solutions and the Stokes phenomenon. Finally, we study hypergeometric equations, one of second order and another of third order, which are related with two generating functions for MZVs, one for 's and another for 's; in particular, we correct a statement from [ZZ3] that the function admits a regular WKB expansion.
Paper Structure (22 sections, 17 theorems, 278 equations)

This paper contains 22 sections, 17 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Some new integral formulas
  3. Generalized hypergeometric series
  4. Confluent hypergeometric series
  5. WKB approximations for confluent hypergeometric functions
  6. Analytic/formal solutions near singular points
  7. Stokes phenomenon
  8. WKB solutions at infinity for completely confluent equation
  9. WKB expansions of hypergeometric functions with a large parameter
  10. Hamilton--Jacobi and transport equations
  11. Testing and principal solutions
  12. Stokes phenomenon
  13. WKB expansions
  14. Variations of hypergeometric functions
  15. Reflections about the Wasow theorem
  16. ...and 7 more sections

Key Result

Theorem 1

Assume that $p=q+1$ and $\mathrm{Re}\beta _{j}>1.$ Then function (2.1) has the following representation: where the integral $\int$ runs over the hypercube $\left\{ 0<\tau _{i}<1:i=1,\ldots ,q\right\}$, the residuum is treated as integration along the $q-$dimension fundamental cycle in the hypersurface of the Gelfand--Leray $q-$form $\mathrm{d}^{q+1}\mathrm{\ln }a/\mathrm{d\ln }\left( a_{1}\cdots a

Theorems & Definitions (34)

  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.1
  • Example 2.2
  • Theorem 2
  • Example 2.3
  • Example 2.4
  • ...and 24 more