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Generating Sequence / Function

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What is a Generating Sequence?

A generating sequence (also called a generating function) is one way to create a finite sequence.

For example:

  • The generating sequence an = cn * rn results in the geometric series if the cns are constant [1].
  • The sequence {6, 26, 66} is generated by the formula [x(x2 + 4x + 1)].

The most important reason for finding the generating function for a sequence is that functions have a much larger “toolbox” to work with. For example, you can’t find derivatives and integrals of sequences, but you can apply those procedures to functions.

Not all sequences have generating sequences. For example, it’s not possible explicitly generate an infinite sequence, but you can generate one for a part of it by using a partial sum [2]. For example, the nth partial sum of the generating sequence an is [3]:
generating sequence

Ordinary Generating Function

The ordinary generating function specifically refers to a formal power series, where the coefficients correspond to a sequence. The general form is:
generating function

Which can also be written as:
G(x) = g0 + g1x + g2x2 + g3x3 + ….

The ordinary generating function is a “formal” power series because the x is used as a placeholder instead of a number; In rare circumstances you might put a value in for x, but leaving the placeholder in means that you can ignore the issue of convergence. [4] It is primarily used in computer science and mathematical analysis.


Other Meanings of “Generating Function”

The term “generating function” is loosely defined. Sometimes it’s used as a synonym for the generating sequence described above. To avoid this confusion, the generating function that specifically generates sequences is sometimes called a function for generating sequences .


Alternatively, it could refer to a specific type of computational tool. For example:

References

[1] Chapter 2: Limits of Sequences.
[2] Basic Calculus Concepts.
[3] Limits of Sequences.
[4] Meyer, A. & Rubinfeld, R. (2005). Generating Functions. Retrieved April 4, 2021 from: https://www.math.cmu.edu/~lohp/docs/math/2011-228/mit-ocw-generating-func.pdf

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Stephanie Glen. "Generating Sequence / Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/generating-sequence-function/
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