## 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 a
_{n}= c_{n}* r^{n}results in the geometric series if the c_{n}s are constant [1]. - The sequence {6, 26, 66} is generated by the formula [x(x
^{2}+ 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 a_{n} is [3]:

## 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:

Which can also be written as:

G(x) = g_{0} + g_{1}x + g_{2}x^{2} + g_{3}x^{3} + ….

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:

- Moment Generating Functions (MGFs) are an alternative way to represent probability distributions; Each distribution has a unique MGF. They are used to find moments like the mean(μ) and variance(σ
^{2}). - Probability Generating Functions are very similar to MGFs and contain the same information. However, the PGF is usually concerned with non-negative, integer-valued random variables.
- A Cumulant Generating Function (CGF) takes the moment of a numerical sequence.

## 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

**CITE THIS AS:**

**Stephanie Glen**. "Generating Sequence / Function" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/generating-sequence-function/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!