International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 59, Pages 3717-3751

On certain Heun functions, associated functions of discrete variables, and applications

P. Malits

Department of Communication Engineering and Center for Applied and Industrial Mathematics at Department of Sciences, Holon Academic Institute of Technology, 52 Golomb Street, Holon 58102, Israel

Received 15 November 2002

Copyright © 2003 P. Malits. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as orthogonal polynomials. Other sets of the coefficients are biorthonormal. Expanding infinite vectors to series with respect to the coefficients is discussed. Certain Legendre functions of complex degree are limiting cases of the studied functions. This leads to new relations for Legendre functions and associated integral transforms. The treated Heun functions find a use for solving dual Fourier series equations which are reduced to the Fredholm integral equations of the second kind. Explicit solutions are obtained in a special case.