  We show that the Heun confluent equation admits infinitely many solutions in
terms of the confluent generalized hypergeometric functions. For each of these
solutions a characteristic exponent of a regular singularity of the Heun
confluent equation is a non-zero integer and the accessory parameter obeys a
polynomial equation. Each of the solutions can be written as a linear
combination with constant coefficients of a finite number of either the Kummer
confluent hypergeometric functions or the Bessel functions.
