4
votes
2answers
71 views

How many numbers are less than million such that their digits sum is $\le 19$?

How many numbers are less than million such that their digits sum is $\le 19$? This question is a Generating-Functions exercise. The solution claims the answer is the coefficient of $x^{19}$ ...
2
votes
1answer
35 views

Why is this function generate $a_n = 2^n(n+1)$?

Let $G(x) = \frac{1}{(1-x)^2}$ which generates the sequence $a_n = n+1$ How can one infer that $G(2x) = \frac{1}{(1-2x)^2}$ generates $a_n = 2^n(n+1)$? Thanks.
2
votes
1answer
42 views

does this sequence of functions converges uniformly to Dirichlet function?

Let $r_{1},r_{2},...$ a sequence that includes all rational numbers in $[0,1]$. Define $$f_n(x)=\begin{cases}1&\text{if }x=r_{1},r_{2},...r_{n}\\0&\text{otherwise}\end{cases}$$ this sequence ...
1
vote
1answer
32 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
2
votes
1answer
84 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
181 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
68 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
104 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
230 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
91 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
184 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
367 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 ...