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The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real valued discrete random variables, the well known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5 TM, allows us to extend to general discrete random variables this technique. Being so, the purpose of this work is twofold. Firstly we show that discretedoi:10.1137/s0040585x97982852 fatcat:dea2qrbkvzcqvops3yre526b2y