[Paper Review] The Appell hypergeometric expansions of the solutions of the general Heun equation

А. М. Ishkhanyan|arXiv (Cornell University)|May 12, 2014

Quantum Mechanics and Non-Hermitian Physics18 references7 citations

TL;DR

This paper presents new series expansions of solutions to the general Heun equation using Appell hypergeometric functions of the first kind in two variables. By analyzing the derivative of the solution, the author derives expansions with coefficients governed by four-term recurrence relations, identifies conditions for termination to finite sums, and constructs the general solution in terms of Gauss hypergeometric functions under specific parameter sets.

ABSTRACT

Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases when the expansions reduce to ones written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain sets of the parameters for which the recurrence relations involve only two terms, though not successive. The coefficients of the expansions are then explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

Motivation & Objective

• To develop new analytical solutions for the general Heun equation using Appell hypergeometric functions.
• To identify parameter conditions under which the series expansions terminate into finite sums.
• To reduce the complexity of coefficient recurrence relations by identifying non-successive two-term relations.
• To express the general solution of the Heun equation in terms of Gauss hypergeometric functions under specific parameter constraints.

Proposed method

• Derive a differential equation for the derivative of the Heun solution to enable expansion in Appell functions.
• Construct series expansions of the Heun solution in terms of Appell F1 hypergeometric functions of two variables.
• Determine recurrence relations for the expansion coefficients, which are generally four-term but reduce to two-term forms under certain parameter sets.
• Identify special parameter configurations where the series reduce to simpler functions such as the incomplete Beta function or Gauss hypergeometric functions.
• Explicitly compute the coefficients of the expansions for cases with two-term recurrence relations.
• Construct the general solution of the Heun equation using Gauss hypergeometric functions when the recurrence relations simplify.

Experimental results

Research questions

• RQ1 Under what conditions do the Appell hypergeometric series expansions of the Heun equation solutions terminate into finite sums?
• RQ2 How do the recurrence relations for the expansion coefficients behave, and when do they reduce to two-term forms (non-successive terms)?
• RQ3 In which parameter regimes do the Appell-based expansions simplify to expressions involving the incomplete Beta function or Gauss hypergeometric functions?
• RQ4 What is the explicit form of the general solution of the Heun equation when the coefficient recurrence relations reduce to two-term forms?
• RQ5 How can the derivative of the Heun solution be used as a starting point to derive new Appell-type expansions?

Key findings

• The coefficients of the Appell hypergeometric expansions obey four-term recurrence relations in the general case.
• Certain parameter sets lead to two-term recurrence relations that are not successive, simplifying the solution structure.
• Finite-sum solutions are obtained when the series terminate, which occurs under specific parameter constraints.
• The general solution of the Heun equation is explicitly constructed in terms of Gauss hypergeometric functions for these special parameter sets.
• The expansions reduce to simpler functions such as the incomplete Beta function when specific parameter conditions are met.
• The method successfully derives new analytical solutions using Appell F1 functions, extending known solution forms for the Heun equation.