How to compute the n-fold convolution power of elementary functions without numerical integration (Q1116272)

Let \(u_ a(t)\) be defined by \(u_ a(t)=1/(2a)\) for \(| t| <a\), \(u_ a(t)=1/(4a)\) for \(| t| =a\), \(u_ a(t)=0\) for \(| t| >a\) and let \(L_ bg(t)=g(t-b)\) the translation of a function g(t) by b. The authors consider density functions f(t) which are elementary functions in the sense that \(f(t)=\sum^{m}_{k=1}c_ KL_{b_ K}u_{a_ K}(t),\) where \(c_ K\) and \(a_ K\) are positive numbers, \(b\in {\mathbb{R}}\) and \(\sum c_ K=1\). A recursion formula is developed which allows to calculate explicitly the n-fold convolution power \(f^{n*}=f*...*f\) of f(t) and the corresponding distribution function \(F_{n*}=\int^{x}_{-\infty}f^{n*}(s)ds.\)