# Tagged Questions

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### 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 ...
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### 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 $G(0)=0$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $0$. ...
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 ...