Generating function: Difference between revisions

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


Let (''a''<sub>''n''</sub>) be a [[sequence]] indexed by the [[natural number]]s.  The ''ordinary generating function'' may be defined purely formally as a [[power series]]
Let (''a''<sub>''n''</sub>) be a [[sequence]] indexed by the [[natural number]]s.  The ''ordinary generating function'' may be defined purely formally as a [[power series]]
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The ''exponential generating function'' may be defined similarly as a power series
The ''exponential generating function'' may be defined similarly as a power series


:<math>A(z) = \sum_{n=0}^\infty \frac{a_n}{n!} z^n .\,</math>
:<math>A(z) = \sum_{n=0}^\infty \frac{a_n}{n!} z^n .\,</math>[[Category:Suggestion Bot Tag]]

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

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

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