# Questions tagged [generating-functions]

Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.

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### Finding a generating function for the Laguerre polynomials

I've started learning some quantum physics and one often encounters special functions (like Legendre polynomials, Laguerre polynomials, Bessel functions, ...). Many calculations with these functions ...
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### Generating function for permutations in $S_n$ with $k$ cycles.

I was reading a little bit about Galois theory, and read that some computer algebra software try to compute Galois groups by finding cycle types. Anyway, this led me to a curious question. If I fix ...
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This is a curiosity I when looking at the binomial theorem. Say you have an ordinary generating function $\displaystyle\sum_{n,m\geq 0}\binom{n}{m}x^ny^m$. This looks kind of like $\displaystyle\sum_{... 4answers 868 views ### Find the ordinary generating function$h(z)$for a Gambler's Ruin variation. Assume we have a random walk starting at 1 with probability of moving left one space$q$, moving right one space$p$, and staying in the same place$r=1-p-q$. Let$T$be the number of steps to reach 0.... 2answers 94 views ### What is the generating function for these word types? I'm curious to see what the generating function is for numbers of some words with a few constraints. Let's fix some$m$, and I'll denote by$[m]$the set of$m$symbols, say$\{1,2,\dots,m\}$. Now ... 1answer 919 views ### The generating function for permutations indexed by number of inversions For$\sigma\in S_n$an inversion is a pair$(\sigma_i,\sigma_j)$such that$i<j$and$\sigma_i>\sigma_j$. Could you help me to prove that the generating function of$S_n$by number of ... 1answer 82 views ### Generating Functions: how do I get my answers in terms of differential operators? I'm reading and enjoying "generatingfunctionology". What a great fun book! But, I'm having some difficulty with the exercises. For example, take the series$a_n = n^2$I'd like to find the Generating ... 4answers 713 views ### How to get closed form from generating function? I have this generating function: $$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{ \frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$ and I know that$\frac {1}{\sqrt {1-4\,z}}$is ... 2answers 199 views ### Solving$A(x) = 2A(x/2) + x^2$Using Generating Functions Suppose I have the recurrence: $$A(x) = 2A(x/2) + x^2$$ with$A(1) = 1$. Is it possible to derive a function using Generating Functions? I know in Generatingfunctionology they shows show to solve ... 2answers 364 views ### how to find generating function with nested sums I'm trying to figure out the generating function for this power series.. I have a few ideas but can't get any result.. $$\sum_{n=2}^\infty \left(\sum_{k=1}^{n} ((n-k)(k-1)M_{k-1}) z^n\right)$$$M(k)...
this is the power series: $$\sum_{i=0}^\infty n(n-1)^2 (n-2) z^n.$$ how can I find a generating function from it? I could use the third derivative but the $n-1$ is squared so I don't know what to do....