Editing Generating function (section)

===Exponential generating function (EGF)===

The ''exponential generating function'' of a sequence {{math|''a''<sub>''n''</sub>}} is
<math display="block">\operatorname{EG}(a_n;x)=\sum_{n=0}^\infty a_n \frac{x^n}{n!}.</math>

Exponential generating functions are generally more convenient than ordinary generating functions for [[combinatorial enumeration]] problems that involve labelled objects.<ref>{{harvnb|Flajolet|Sedgewick|2009|p=95}}</ref>

Another benefit of exponential generating functions is that they are useful in transferring linear [[recurrence relations]] to the realm of [[differential equations]]. For example, take the [[Fibonacci sequence]] {{math|{''f<sub>n</sub>''}<nowiki/>}} that satisfies the linear recurrence relation {{math|''f''<sub>''n''+2</sub> {{=}} ''f''<sub>''n''+1</sub> + ''f''<sub>''n''</sub>}}. The corresponding exponential generating function has the form
<math display="block">\operatorname{EF}(x) = \sum_{n=0}^\infty \frac{f_n}{n!} x^n</math>

and its derivatives can readily be shown to satisfy the differential equation {{math|EF{{pprime}}(''x'') {{=}} EF{{prime}}(''x'') + EF(''x'')}} as a direct analogue with the recurrence relation above. In this view, the factorial term {{math|''n''!}} is merely a counter-term to normalise the derivative operator acting on {{math|''x''<sup>''n''</sup>}}.