In this paper, we introduce new functions as generalizations of the incomplete gamma functions. The functions are found to be useful in heat conduction, probability theory and in the study of Fourier and Laplace transforms. Some important properties of the functions are studied. We have investigated the asymptotic behavior, Laplace transforms, special cases, decomposition formula, integral representations, convolutions, recurrence relations and differentiation formula of these functions. Applications of these functions in evaluation of certain inverse Laplace transforms to the definite integrals and to the infinite series of exponential functions are shown.
