Editing Generating function (section)

==== Multiplication yields convolution ====
{{Main|Cauchy product}}

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]])
<math display="block">(a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots)</math>
of a sequence with ordinary generating function {{math|''G''(''a<sub>n</sub>''; ''x'')}} has the generating function
<math display="block">G(a_n; x) \cdot \frac{1}{1-x}</math>
because {{math|{{sfrac|1|1 − ''x''}}}} is the ordinary generating function for the sequence {{nowrap|(1, 1, ...)}}. See also the [[#Convolution (Cauchy products)|section on convolutions]] in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.