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Asymptotic expansions of the Humbert Function $Φ_1$ and their applications

Peng-Cheng Hang, Liangjian Hu, Min-Jie Luo

Abstract

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $Φ_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii) $y\to\infty$; (iii) $x\to\infty,\,y\to\infty$; (iv) $x$ or $y$ small, $xy$ fixed; and (v) $x\to 1$, $y$ fixed. The utility of these expansions is illustrated through concrete applications in the theory of Saran's hypergeometric function $F_M$, the Glauber-Ising model, and the theory of Prabhakar-type fractional integral operators. Several potential directions for future work are also outlined.

Asymptotic expansions of the Humbert Function $Φ_1$ and their applications

Abstract

This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function . Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) ; (ii) ; (iii) ; (iv) or small, fixed; and (v) , fixed. The utility of these expansions is illustrated through concrete applications in the theory of Saran's hypergeometric function , the Glauber-Ising model, and the theory of Prabhakar-type fractional integral operators. Several potential directions for future work are also outlined.

Paper Structure

This paper contains 14 sections, 21 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Regime (i):
  5. Regime (ii):
  6. Regime (iii):
  7. Regime (iv):
  8. Regime (v):
  9. Applications
  10. Analytic continuations of
  11. Glauber-Ising model
  12. Integral transforms involving
  13. Discussion
  14. Proof of Eq.

Key Result

Theorem 2.1

Assume that $a,b\in\mathbb{C},\,c\in\mathbb{C}\setminus\mathbb{Z}_{\leqslant 0}$ and $a+b-c\notin\mathbb{Z}$. Then for $(x,y)\in\mathbb{D}$,

Theorems & Definitions (39)

  • Theorem 2.1: Tuan-Kalla 1987
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Remark 1
  • Theorem 3.2
  • proof
  • Remark 2
  • Corollary 3.3
  • Remark 3
  • ...and 29 more