2
votes
1answer
75 views

Differentiate $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+…+z^{i-1}}{i}$ twice to calculate the variance of involutions.

Use the Probability Generating Function for Involutions: $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+...+z^{i-1}}{i}$ To Calculate the Variance of Involutions where: $Variance \space X_n = ...
8
votes
2answers
164 views

Ordinary generating function for $\binom{3n}{n}$

The ordinary generating function for the central binomial coefficients, that is, $$\displaystyle \sum_{n=0}^{\infty} \binom{2n}{n} x^{n} = \frac{1}{\sqrt{1-4x}}$$ follows from the generalized ...
1
vote
1answer
65 views

moment generating function of gambling [duplicate]

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1-p$. Let $f_n$ be the probability that he or ...
4
votes
1answer
94 views

Inversion of a power series without a linear term

Could someone explain me how to invert $$ z = y e^{-y} = e^{-1} - \frac{1}{2e}(y - 1)^2 + \frac{1}{3e}(y - 1)^3 - \frac{1}{8e}(y - 1)^4 + \cdots $$ around $y=1, z=e^{-1}$, so that $y$ is expressed as ...
1
vote
1answer
204 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
0
votes
1answer
89 views

How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?

In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or ...
3
votes
1answer
177 views

Bernoulli and Euler numbers in some known series.

The series for some day to day functions such as $\tan z$ and $\cot z$ involve them. So does the series for $\dfrac{z}{e^z-1}$ and the Euler Maclaurin summation formula. How can it be analitically ...
1
vote
1answer
347 views

Advanced application of the Binomial Theorem

I'm trying to solve the following integral: $$ \int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx $$ Where $C_{n}^{\lambda}(x)$ is a ...