Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Non-linear integral equations for Heun functions

B. D. Sleemana1

a1 Department of Mathematics, The University, Dundee

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the form

S0013091500012943_eqn1

Further, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

(Received August 14 1968)