  L\'{e}vy processes with completely monotone jumps appear frequently in
various applications of probability. For example, all popular stock price
models based on L\'{e}vy processes (such as the Variance Gamma, CGMY/KoBoL and
Normal Inverse Gaussian) belong to this class. In this paper we continue the
work started in [Int. J. Theor. Appl. Finance 13 (2010) 63-91, Quant. Finance
10 (2010) 629-644] and develop a simple yet very efficient method for
approximating processes with completely monotone jumps by processes with
hyperexponential jumps, the latter being the most convenient class for
performing numerical computations. Our approach is based on connecting L\'{e}vy
processes with completely monotone jumps with several areas of classical
analysis, including Pad\'{e} approximations, Gaussian quadrature and orthogonal
polynomials.
