  The paper is an investigation of the analytic properties of a new class of
special functions that appear in the kernels of a class of integral operators
underlying the dynamics of matter relaxation processes in attractive fields.
These functions, recently introduced by the second author, generate the kernels
of the principal parts of these operators and play an important role in
understanding their spectral characteristics. We reveal the representations of
these functions in terms of the Gauss and Clausen hypergeometric functions and
present differential-difference and differential equations they satisfy.
Mathematically, the results include calculation of certain trigonometric double
integrals and derivation of their other properties. Furthermore, they represent
a potentially useful tool in matter relaxation in an external field, the study
of nanoelectronic electrolyte-based systems and dynamics of charge carriers in
media with obstacles.
