FindGeneratingFunction [{a0,a1,…},x]
attempts to find a simple generating function in x whose n^(th) series coefficient is an.
FindGeneratingFunction [{{n0,a0},{n1,a1},…},x]
attempts to find a simple generating function whose ni^(th) series coefficient is ai.
FindGeneratingFunction
FindGeneratingFunction [{a0,a1,…},x]
attempts to find a simple generating function in x whose n^(th) series coefficient is an.
FindGeneratingFunction [{{n0,a0},{n1,a1},…},x]
attempts to find a simple generating function whose ni^(th) series coefficient is ai.
Details and Options
- The sequence elements an 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:
-
TimeConstraint 10 how many seconds to search a particular function space or perform a transformation
- 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.
Examples
open all close allBasic Examples (2)
Find a generating function for a sequence:
A periodic sequence:
Scope (2)
Rational functions:
Hypergeometric functions:
Generalizations & Extensions (1)
FindGeneratingFunction works on arbitrary exact numbers or symbolic expressions:
Properties & Relations (1)
Use FindSequenceFunction to find a generating function of a sequence:
Verify:
See Also
FindSequenceFunction GeneratingFunction DifferentialRoot Series SeriesCoefficient FindLinearRecurrence
Function Repository: RecognizeSeries
Related Guides
Related Links
History
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.html
BibTeX
@misc{reference.wolfram_2025_findgeneratingfunction, author="Wolfram Research", title="{FindGeneratingFunction}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FindGeneratingFunction.html}", note=[Accessed: 16-November-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: 16-November-2025]}