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FindGeneratingFunction—Wolfram Language Documentation

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BUILT-IN SYMBOL

FindGeneratingFunction[{a₀,a₁,…},x]

attempts to find a simple generating function in x whose n series coefficient is a_n.

FindGeneratingFunction[{{n₀,a₀},{n₁,a₁},…},x]

attempts to find a simple generating function whose n_i series coefficient is a_i.

Details and Options

The sequence elements a_n can be either exact numbers or symbolic expressions.
FindGeneratingFunction finds results in terms of a wide range of integer functions, as well as implicit solutions to difference equations represented by DifferenceRoot.
If FindGeneratingFunction cannot find a simple generating function that yields the specified sequence, it returns unevaluated.
FindGeneratingFunction has the following options:
FindGeneratingFunction[list,x] by default uses earlier elements in list to find candidate simple generating functions, then validates the generating functions by looking at later elements.
FindGeneratingFunction[list,x] only returns functions that correctly reproduce all elements of list.

Examplesopen allclose all Basic Examples (2)

Find a generating function for a sequence:

A periodic sequence:

Scope (2)

Rational functions:

Hypergeometric functions:

Generalizations & Extensions (1) Properties & Relations (1) Wolfram Research (2008), FindGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindGeneratingFunction.html. Text Wolfram Research (2008), FindGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindGeneratingFunction.html.CMS Wolfram Language. 2008. "FindGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FindGeneratingFunction.html.APA Wolfram Language. (2008). FindGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindGeneratingFunction.htmlBibTeX @misc{reference.wolfram_2025_findgeneratingfunction, author="Wolfram Research", title="{FindGeneratingFunction}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FindGeneratingFunction.html}", note=[Accessed: 12-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_findgeneratingfunction, organization={Wolfram Research}, title={FindGeneratingFunction}, year={2008}, url={https://reference.wolfram.com/language/ref/FindGeneratingFunction.html}, note=[Accessed: 12-July-2025 ]}

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