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
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 (-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.