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1	Generating Functions : Powerful Tools for Recurrence Relations. Hermite Polynomials Generating Function Rydén, Christoffer January 2023 (has links) In this report we will plunge down in the fascinating world of the generating functions. Generating functions showcase the "power of power series", giving more depth to the word "power" in power series. We start off small to get a good understanding of the generating function and what it does. Also, off course, explaining why it works and why we can do some of the things we do with them. We will see alot of examples throughout the text that helps the reader to grasp the mathematical object that is the generating function. We will look at several kinds of generating functions, the main focus when we establish our understanding of these will be the "ordinary power series" generating function ("ops") that we discuss before moving on to the "exponential generating function" ("egf"). During our discussion on ops we will see a "first time in literature" derivation of the generating function for a recurrence relation regarding "branched coverings". After finishing the discussion regarding egf we move on the Hermite polynomials and show how we derive their generating function. Which is a generating function that generates functions. Lastly we will have a quick look at the "moment generating function". Exponential generating function Moment generating function Hermite polynomials Mathematics Matematik
2	Applications of Generating Functions Tseng, Chieh-Mei 26 June 2007 (has links) Generating functions express a sequence as coefficients arising from a power series in variables. They have many applications in combinatorics and probability. In this paper, we will investigate the important properties of four kinds of generating functions in one variables: ordinary generating unction, exponential generating function, probability generating function and moment generating function. Many examples with applications in combinatorics and probability, will be discussed. Finally, some well-known contest problems related to generating functions will be addressed. moment generating function ordinary generating function exponential generating function probability generating function recurrence relation recurrence binomial coefficient binomial theorem weak law of large numbers central limit theorem moment probability distribution variance power series identities induction counting problem partition Fibonacci partial fractions sequence expectation convolution

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Generating Functions : Powerful Tools for Recurrence Relations. Hermite Polynomials Generating Function

Applications of Generating Functions