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Multiplication yields convolution

Multiplication of ordinary generating functions yields a discrete convolution (the Cauchy product) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general Euler–Maclaurin formula) (a 0, a 0 + a 1, a 0 + a 1 + a 2, …) {\displaystyle (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots)} of a sequence with ordinary generating function G (a; x) has the generating function G (a n; x) ⋅ 1 1 − x {\displaystyle G(a_{n};x)\cdot {\frac {1}{1-x}}} because 1 / 1 − x is the ordinary generating function for the sequence (1, 1,...). See also the section on convolutions in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.