Electron. J. Diff. Eqns., Vol. 2007(2007), No. 94, pp. 1-11.

Integral representation of a solution of Heun's general equation

François Batola, Jomo Batola

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.

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François Batola
Centre de Recherche Mathématique et Physique d'Avensan (CRMPA)
8 Chemin de Loze, 33480 Avensan, France
email: crmpa@wanadoo.fr
Jomo Batola
Faculty of Technology
Southampton Solent University, SO14 0RD, UK
email: crmpa@wanadoo.fr

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