1
vote
1answer
29 views

Number of ways distribute 12 identical action figures to 5 children

Need a little help with this problem. Use generating functions to determine the number of different ways 12 identical action figures can be given to five children so that each child receives at most ...
0
votes
1answer
58 views

Convergence of formal power series substitution

Prove that the substitution of formal power series $F(G(x))=\sum_{k\geq0}f_k \frac{G(x)^k}{n!}$ converges for every $F$ if and only if $G(0)=0$
1
vote
1answer
75 views

Question about the infinite products of formal power series

I need a proof for this: Let $(F_j)_{j\ge 0}$ be a sequence of formal power series. The infinite product $\prod_{j\geq0}(1+F_j(x))$, where $F_j(0)=0$, converges if and only if ...
3
votes
2answers
340 views

Formal power series, the Chain Rule and the Product Rule.

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$. Exercise i) If $F_1(x)$ and $F_2(x)$ are power series ...
2
votes
0answers
91 views

Simplify the square of a sum of cosine functions

I have a square sum of exponantials as below: $$\left|\sum_{l=0}^{M-1}\exp\left(jl^2a\right)\,\exp\left(\frac{-j2\pi l}{M}b\right)\right|^2 $$ where $a$ is constant and $b$ is an integer . and I have ...
0
votes
1answer
225 views

find the power series of the problem…

""Find a power series associated with the problem where we have to find a number of ways to select 10 people to form and expert committee from 6 Professors and 12 Associate Professors."" Question ...
2
votes
3answers
206 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
3
votes
4answers
218 views

Transforming power series

Definition: Let $$a_0 = a_1 = 1, \; a_{n+2} = a_{n+1} + (n+1) \cdot a_n, \; n \geq 0$$ Exercise: Prove that $$\sum_{n\geq 0} \frac{a_n}{n!} x^n = \exp \left( x + \frac{1}{2} x^2 \right)$$ I ...
4
votes
2answers
498 views

Proofs for formal power series

Definitons Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n : a_n \in \mathbb{C} \right\}$$ be the set of formal power series of x and $$F(x) = \sum_{n\geq 0} a_n x^n, \; G(x) = \sum_{n\geq 0} ...
4
votes
2answers
619 views

Turning a closed-form generating function back to ordinary power series

If I know the formal power series, I know how to find the closed form: $$\displaystyle F = \sum_{n=0}^{\infty} {X^n} = 1 + X^1 + X^2 + X^3 + ...$$ $$\displaystyle F \cdot X = X \cdot ...