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:
  • FunctionSpaceAutomaticwhere to look for candidate simple generating functions
    MethodAutomaticmethod to use
    TimeConstraint10how many seconds to search a particular function space or perform a transformation
    ValidationLengthAutomaticsequence length used to validate a candidate generating function found
  • 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

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Basic 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:

Introduced in 2008
 (7.0)