[Paper Review] New transformation law for Heun and Hypergeometric Equations

Yves Gaspar|ArXiv.org|May 23, 2003

Numerical methods for differential equations3 references3 citations

TL;DR

This paper introduces new non-linear transformations between Heun and hypergeometric differential equations by redefining the independent variable, enabling mappings between Fuchsian and non-Fuchsian equations. The key contribution is a novel Heun-to-Hypergeometric transformation that generalizes known results by allowing non-linear variable changes, revealing new connections between equations with different singularity structures.

ABSTRACT

In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some simple examples of transformations between equations that are not Fuchsian and that generalise the Heun-to-Hypergeometric transformations.

Motivation & Objective

• To develop new Heun-to-Heun and Heun-to-Hypergeometric transformations using non-linear changes of the independent variable.
• To generalize existing hypergeometric identities by extending them beyond linear variable transformations.
• To explore transformations between equations that are not Fuchsian, particularly those with mixed regular and irregular singularities.
• To establish a framework connecting Heun-type equations with hypergeometric-type equations through non-linear Riccati-based mappings.
• To demonstrate that non-linear transformations can relate equations with different numbers of regular singularities, including cases with one irregular singularity.

Proposed method

• Utilizes a general Riccati transformation framework linking solutions of two second-order linear ODEs via the relation $ (dy/dx)/y \cdot (du/dx)/u = f(x) $.
• Applies non-linear variable transformations $ \xi = \lambda / (1 - c_2(x + \mu)) $ to map solutions of the Heun equation to new forms.
• Derives transformed equations by expressing $ F_1(\xi) $ and $ \alpha(\xi) $ as functions of $ x $, enabling transformation of the Heun equation into a hypergeometric-type equation.
• Imposes conditions such as $ N(x) = R(mx + n)^2 $ to reduce the number of regular singularities from four to three, enabling Heun-to-Hypergeometric reduction.
• Solves a system of algebraic equations (24)–(26) to determine parameters $ c_1, \mu, R $ that satisfy the singularity reduction condition.
• Uses the transformation $ h(x) = (dk/d\xi)/k \cdot (d\xi/dx) $ to relate the logarithmic derivative of the solution to the new equation, preserving the structure of the original ODE.

Experimental results

Research questions

• RQ1 Can non-linear transformations of the independent variable yield new Heun-to-Heun and Heun-to-Hypergeometric transformations?
• RQ2 How do non-linear variable changes affect the singularity structure of second-order linear ODEs?
• RQ3 Can transformations be constructed that relate non-Fuchsian equations (with irregular singularities) to Fuchsian ones?
• RQ4 What algebraic conditions must parameters satisfy for a Heun equation to be transformed into a hypergeometric-type equation?
• RQ5 How does the Riccati-based transformation framework extend beyond linear variable changes to include non-linear mappings?

Key findings

• A new Heun-to-Hypergeometric transformation is established using a non-linear change of the independent variable, generalizing previous results that relied on linear or unchanged variables.
• The transformation reduces a Heun equation with four regular singularities to a hypergeometric-type equation with three regular singularities by imposing $ N(x) = R(mx + n)^2 $.
• The parameter $ c_1 $ is determined as $ c_1 = (a - (c-1)m)/(R m^2) $, linking the solution structure to the transformation parameters.
• The parameter $ \mu $ is derived as $ \mu = [(a - (c-1)m)(2mnR + m) + (-b + (c-1)n)R m^2] / [(a - (c-1)m)R m^2] $, ensuring the singularity condition is met.
• The parameter $ R $ is explicitly solved as $ R = (L^2 n - (a m - l m^2) K) / ((2a m n - b m^2 - l m^2 n)K - L^2 n^2) $, with $ l = c-1 $, $ L = a - l m $, $ K = b - n l $, enabling full parameterization of the transformation.
• The method reveals that even when $ \epsilon \neq 0 $, a Heun equation with four regular singularities can be mapped to a hypergeometric-type equation, demonstrating a generalization beyond the standard Fuchsian case.