Generating function
Exponential generating function (EGF)

The exponential generating function of a sequence a is EG (a n; x) = ∑ n = 0 ∞ a n x n n !. {\displaystyle \operatorname {EG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}.}

Exponential generating functions are generally more convenient than ordinary generating functions for combinatorial enumeration problems that involve labelled objects.

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 { f } that satisfies the linear recurrence relation f = f + f. The corresponding exponential generating function has the form EF (x) = ∑ n = 0 ∞ f n n ! x n {\displaystyle \operatorname {EF} (x)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}x^{n}}

and its derivatives can readily be shown to satisfy the differential equation EF″(x) = EF ′ (x) + EF(x) as a direct analogue with the recurrence relation above. In this view, the factorial term n ! is merely a counter-term to normalise the derivative operator acting on x.
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