generating function
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English
Etymology
The concept was introduced by French mathematician Abraham de Moivre in 1730.
Noun
generating function (plural generating functions)
- (mathematics) A formal power series with one indeterminate, whose coefficients encode a sequence that can be studied by algebraic manipulation of the series; any one of several generalizations, such as to encode more than one sequence or use more than one indeterminate.
- 1954, George Pólya, Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics, Princeton University Press, page 101:
- A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.
- 1990, Herbert S. Wilf, generatingfunctionology, Academic Press, page 2:
- Most often generating functions arise from recurrence formulas. Sometimes, however, from a generating function you will find a new recurrence formula, not the one you started with, that gives new insights into the nature of your sequence.
- 2003, Sergei K. Lando (author & translator), Lectures on Generating Functions, American Mathematical Society.
Usage notes
Despite the name, a generating function is not a function. As a formal power series, it is understood that its indeterminate (not a "variable") is never assigned a value and the series is never evaluated. In fact, the series is not even required to converge.
Hypernyms
Hyponyms
- exponential generating function
- Bell series
- Dirichlet series
- Lambert series
- ordinary generating function
Translations
formal power series whose coefficients encode a sequence
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See also
- moment-generating function (probability, statistics)
- power series
Further reading
Examples of generating functions on Wikipedia.Wikipedia - Polynomial sequence on Wikipedia.Wikipedia
- Generating function transformation on Wikipedia.Wikipedia
