Abstract
This study investigates both the homogeneous and non-homogeneous forms of the
k
-confluent hypergeometric differential equation. In the neighborhood of the regular singular point, series solutions are constructed using the Frobenius method. For the irregular singular point at infinity, an asymptotic analysis is carried out to describe the behavior of the solutions in that region. These analyses lead to the definition and characterization of
k
-Kummer and
k
-Tricomi-type hypergeometric functions, which are shown to constitute fundamental solutions of the differential equation. Moreover, an explicit series representation is obtained for the non-homogeneous
k
-confluent hypergeometric differential equation with a polynomial right-hand side. The results provide analytical tools for obtaining explicit and asymptotic solutions of differential equations of confluent hypergeometric type, including models arising in wave propagation, heat conduction, and quantum mechanics.