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15: Generating Functions

  • Page ID
    • Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
    • Google and Massachusetts Institute of Technology via MIT OpenCourseWare
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    Generating Functions are one of the most surprising and useful inventions in Discrete Mathematics. Roughly speaking, generating functions transform problems about sequences into problems about algebra. This is great because we’ve got piles of algebraic rules. Thanks to generating functions, we can reduce problems about sequences to checking properties of algebraic expressions. This will allow us to use generating functions to solve all sorts of counting problems.

    Several flavors of generating functions such as ordinary, exponential, and Dirichlet come up regularly in combinatorial mathematics. In addition, Z-transforms, which are closely related to ordinary generating functions, are important in control theory and signal processing. But ordinary generating functions are enough to illustrate the power of the idea, so we’ll stick to them. So from now on generating function will mean the ordinary kind, and we will offer a taste of this large subject by showing how generating functions can be used to solve certain kinds of counting problems and how they can be used to find simple formulas for linear-recursive functions.

    This page titled 15: Generating Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer (MIT OpenCourseWare) .

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