0
votes
1answer
56 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
74 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
338 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
494 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
614 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 ...