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# Exercises of Chapter 1 of gfology

1. Find the ordinary power series generating functions of each of the following sequences, in simple, closed-form. In each case, the sequence is defined for all $$n\ge 0$$.
• $$a_n=n$$.

sol:

\begin{aligned} f(z)&=\sum_{n\ge 0} nz^n=z\sum_{n\ge 1}nz^{n-1}\\ &=z\sum_{n\ge 1}(z^n)'=z\left(\sum_{n\ge 1}z^n\right)'\\ &=\dfrac{z}{(1-z)^2} \end{aligned}

• $$a_n=\alpha n+\beta$$.

sol:

\begin{aligned} f(z)&=\sum_{n\ge 0} (\alpha n+\beta)z^n\\ &=\alpha\dfrac{\alpha}{(1-z)^2}+\dfrac{1}{(1-z)} \end{aligned}

• $$a_n=3^n$$

sol:

\begin{aligned} f(z)=\sum_{n\ge 0} 3^nz^n=\sum_{n\ge 0} (3z)^n = \dfrac{1}{1-3z} \end{aligned}

1. For each of the sequences given in part 1, find the exponential generating function of the sequence in simple, closed-form.
• $$a_n=n$$.

sol:

\begin{aligned} f(z)=\sum_{n\ge 1}\dfrac{z^{n}}{(n-1)!}=z\sum_{n\ge 0}\dfrac{z^n}{n!}=ze^z \end{aligned}

• $$a_n=\alpha n+\beta$$

sol:

$f(z)=\alpha\sum_{n\ge 0} n\dfrac{z^n}{n!}+\beta\sum_{n\ge 0} \dfrac{z^n}{n!}=(\alpha z+\beta)e^z$

• $$a_n=3^n$$

sol:

$f(z)=\sum_{n\ge 0}\dfrac{(3z)^n}{n!}=e^{3z}$

1. If $$f(z)$$ is the ordinary power series generating function of the sequence $$\{a_n\}_{n\ge 0}$$, then express simply in terms of $$f(z)$$, the ordinary power series generating functions of the following sequences. In each case the range of $$n$$ is $$0,1,2,\dots$$
• $$\{a_n+c\}$$
sol:

$g(z)=\sum_{n\ge 0}(a_n+c)z^n=f(z)+\dfrac{c}{(1-z)}$

• $$\{na_n\}$$

sol:

$g(z)=\sum_{n\ge 0} na_nz^n=z\sum_{n\ge 1}na_nz^{n-1}=\sum_{n\ge 1}a_n(z^n)'=(f(z)-a_0)'=f'(z)$

• $$\{a_{n+h}\}$$, $$h$$ is a given constant.

$g_z=\sum_{n\ge 0}a_{n+h}z^n=\dfrac{f(z)-\sum_{j=0}^{h-1} a_jz^j}{z^h}$

1. Find
• $$[z^n] e^{2z}$$

sol:

$[z^n] e^{2z}=\dfrac{2^n}{n!}$

• $$\left[\dfrac{z^n}{n!}\right]e^{\alpha z}$$

sol:

$\left[\dfrac{z^n}{n!}\right]e^{\alpha z}=\alpha^n$

• $$[z^n/n!]\sin z$$

sol:

$( [z^n/n!]\sin z=[z^n/n!]\dfrac{e^{iz}-e^{-iz}}{2i}=\dfrac{[z^n/n!](e^{iz}-e^{-iz})}{2i}=\dfrac{1+(-1)^{n+1}}{2}$

posted @ 2022-03-28 10:30  feicheng  阅读(10)  评论(0编辑  收藏  举报