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.