Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 94, pp. 1-11.

Title: Integral representation of a solution of Heun's general equation

Authors: Francois Batola (CRMPA, Avensan,  France)
         Jomo Batola (Southampton Solent Univ., UK)

Abstract:
 We establish an integral representation for the Frobenius solution
 with an exponent zero at $z=0$ of the general Heun equation.
 First we present an extension of Mellin's lemma which provides
 a powerful method that takes into account differential equations
 which are not of the form studied by Mellin. That is the case
 for equations of Heun's type.
 It is that aspect which makes our work different from  Valent's work.
 The method is powerful because it allows obtaining directly the nucleus
 equation of the representation. The integral representation formula
 obtained with this method leads quickly and naturally to already known
 results in the case of hypergeometric functions.
 The generalisation of this method gives a type of differential equations
 which form is a novelty and deserves to be studied further.


Submitted November 16, 2006. Published June 21, 2007.
Math Subject Classifications: 30B40, 30D10, 33E20, 33E30; 33C05, 34M05.
Key Words: Heun equation; Heun function; integral representation;
           analytic continuation; extension of Mellin lemma; 
	   integral relation.