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First Derivative of Automorphic Function of Triangle Groups

Md. Shafiul Alam, Bijan Krishna Saha, Chinmayee Podder

TL;DR

This paper derives an explicit first derivative formula for the G-automorphic function associated with a triangle group in terms of Gaussian hypergeometric functions. It uses the Schwarz triangle function as the inverse of the hypergeometric differential equation and a hypergeometric based mapping to the hyperbolic triangle to obtain a tractable expression. The key contribution is the closed form for dξ/dw with parameters α,β,γ and α',β',γ' determined by the triangle group data and an accessory parameter K defined via Gamma functions, together with a hyperbolic distance lemma and a detailed Wronskian Abel-based proof. The results provide a computable representation of automorphic functions on triangle groups in terms of classical special functions and illuminate their hyperbolic geometric structure.

Abstract

For a triangle group $G$, the $G$-automorphic function is the inverse of Schwarz triangle function. In this paper, we compute the first derivative of the $G$-automorphic function for the triangle group $G$ in terms of the Gaussian hypergeometric function.

First Derivative of Automorphic Function of Triangle Groups

TL;DR

This paper derives an explicit first derivative formula for the G-automorphic function associated with a triangle group in terms of Gaussian hypergeometric functions. It uses the Schwarz triangle function as the inverse of the hypergeometric differential equation and a hypergeometric based mapping to the hyperbolic triangle to obtain a tractable expression. The key contribution is the closed form for dξ/dw with parameters α,β,γ and α',β',γ' determined by the triangle group data and an accessory parameter K defined via Gamma functions, together with a hyperbolic distance lemma and a detailed Wronskian Abel-based proof. The results provide a computable representation of automorphic functions on triangle groups in terms of classical special functions and illuminate their hyperbolic geometric structure.

Abstract

For a triangle group , the -automorphic function is the inverse of Schwarz triangle function. In this paper, we compute the first derivative of the -automorphic function for the triangle group in terms of the Gaussian hypergeometric function.
Paper Structure (3 sections, 4 theorems, 51 equations)

This paper contains 3 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Main Results and Proofs
  3. Proof of Theorem \ref{['thm1']}.

Key Result

Theorem A

For $\gamma \ne 1$, let then the functions are linearly independent solutions of (hyp). Let $w=f(\xi)=\frac{g_2(\xi)}{g_1(\xi)}$, then $\mathcal{H}=\{\xi\in\mathbb{C}: {\operatorname{Im}\,} \xi>0\}$ is mapped by $f(\xi)$ conformally to the hyperbolic (non-Euclidean) triangle $[w_1, w_2, w_3]$ in the $w$-plane, where The interior angles at the vertices $w_1$, $w_2$ and $w_3$ are, respectively, $

Theorems & Definitions (6)

  • Theorem A: $bayer$
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.2.1
  • proof