Generating function

In mathematics, a generating function is a function for which the definition "encodes" values of a sequence, allowing the application of methods of real and complex analysis to problems in algorithmics, combinatorics, number theory, probability and other areas. It is the basis of the engineering term function generator, devices which produce periodic or aperiodic signals when physical and software parameters, based on mathematical terminology, are entered.

Let (a_n) be a sequence indexed by the natural numbers. The ordinary generating function may be defined purely formally as a power series

$A(z)=\sum _{n=0}^{\infty }a_{n}z^{n},\,$

where for the present we do not address issues of convergence.

The exponential generating function may be defined similarly as a power series

$A(z)=\sum _{n=0}^{\infty }{\frac {a_{n}}{n!}}z^{n}.\,$