**Infinitely Divisible 2-EPT Distributions and their Levy Processes**

The class of probability density functions on

**R**with strictly proper rational characteristic functions are considered. On [0, ∞) as well as (-∞, 0) these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions which we abbreviate with as 2-EPT densities.EPT density functions can be represented as f(x)=**c**e^{Ax}**b**, where**A**is a square matrix,**b**a column vector and**c**a row vector. The triple (**A, b, c**) is called the minimal realization of the EPT density function. The class of probability measures on**R**with (proper) rational characteristic functions is also considered whose densities correspond to mixtures of the pointmass at zero ("delta distribution") and 2-EPT densities. These mixture distributions are referred to as generalised 2-EPT probability density functions. A sufficient and necessary condition is derived to characterise infinitely divisible EPT and generalised EPT probability density functions. Following a similar procedure, an analogous necessary and sufficient condition is derived to characterise infinitely divisible rational functions, defined on the half line, using the locations, including multiplicities, of its poles and zeros. It is then proven that an infinitely divisible 2-EPT probability density function is the convolution of two infinitely divisible EPT functions, defined on [0, ∞) and (-∞, 0] respectively. Hence, it follows that to characterising an infinitely divisible 2-EPT density function is equivalent to characterising two infinitely divisible EPT functions, after the Laplace transform of the 2-EPT function has been appropriately factorised. The Levy processes generated by infinitely divisible EPT and 2-EPT distributions are seen to be of finite variation. The Levy triples of such processes are also computed.