Mixing
Sequence $rightarrow$ Generating function simple practice problems
$begingroup$
I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some practice problems that you think would be helpful for someone just learning about generating functions? Thanks.
For the sake of context, an example.
Let $a,b,r_0,r_1$ be fixed real numbers. Then define for $ninBbb N_0$,
$$r_{n+2}=ar_{n+1}+br_n$$
and $$R(x)=sum_{ngeq0}r_nx^n$$
Multiplying both sides of the recurrence by $x^{n+2}$ and summing from $n=0$ to $infty$,
$$R(x)-r_0-r_1x=axR(x)-ar_0x+bx^2R(x)$$
$$R(x)=frac{r_0+(r_1-ar_0)x}{1-ax-bx^2}$$
Currently the most advanced thing like this I've ever done is show that
$$sum_{ngeq1}H_{n}^{(k)}s^n=frac{mathrm{Li}_k(s)}{1-s}$$
Where $$H_n^{(k)}:=sum_{i=1}^{n}frac1{i^k}$$ and $$mathrm{Li}_k(s):=sum_{ngeq1}frac{s^n}{n^k}$$ Are respectively the harmonic numbers of order $k$, and the $k$-th polylogarithm function.
Hit me!
soft-question recurrence-relations generating-functions big-list
asked Jan 17 at 5:28
clathratusclathratus
4,640337
$endgroup$
add a comment |
$begingroup$
I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some practice problems that you think would be helpful for someone just learning about generating functions? Thanks.
For the sake of context, an example.
Let $a,b,r_0,r_1$ be fixed real numbers. Then define for $ninBbb N_0$,
$$r_{n+2}=ar_{n+1}+br_n$$
and $$R(x)=sum_{ngeq0}r_nx^n$$
Multiplying both sides of the recurrence by $x^{n+2}$ and summing from $n=0$ to $infty$,
$$R(x)-r_0-r_1x=axR(x)-ar_0x+bx^2R(x)$$
$$R(x)=frac{r_0+(r_1-ar_0)x}{1-ax-bx^2}$$
Currently the most advanced thing like this I've ever done is show that
$$sum_{ngeq1}H_{n}^{(k)}s^n=frac{mathrm{Li}_k(s)}{1-s}$$
Where $$H_n^{(k)}:=sum_{i=1}^{n}frac1{i^k}$$ and $$mathrm{Li}_k(s):=sum_{ngeq1}frac{s^n}{n^k}$$ Are respectively the harmonic numbers of order $k$, and the $k$-th polylogarithm function.
Hit me!
soft-question recurrence-relations generating-functions big-list
asked Jan 17 at 5:28
clathratusclathratus
4,640337
$endgroup$
1
$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35
add a comment |
$begingroup$
I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some practice problems that you think would be helpful for someone just learning about generating functions? Thanks.
For the sake of context, an example.
Let $a,b,r_0,r_1$ be fixed real numbers. Then define for $ninBbb N_0$,
$$r_{n+2}=ar_{n+1}+br_n$$
and $$R(x)=sum_{ngeq0}r_nx^n$$
Multiplying both sides of the recurrence by $x^{n+2}$ and summing from $n=0$ to $infty$,
$$R(x)-r_0-r_1x=axR(x)-ar_0x+bx^2R(x)$$
$$R(x)=frac{r_0+(r_1-ar_0)x}{1-ax-bx^2}$$
Currently the most advanced thing like this I've ever done is show that
$$sum_{ngeq1}H_{n}^{(k)}s^n=frac{mathrm{Li}_k(s)}{1-s}$$
Where $$H_n^{(k)}:=sum_{i=1}^{n}frac1{i^k}$$ and $$mathrm{Li}_k(s):=sum_{ngeq1}frac{s^n}{n^k}$$ Are respectively the harmonic numbers of order $k$, and the $k$-th polylogarithm function.
Hit me!
soft-question recurrence-relations generating-functions big-list
asked Jan 17 at 5:28
clathratusclathratus
4,640337
$endgroup$
I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some practice problems that you think would be helpful for someone just learning about generating functions? Thanks.
For the sake of context, an example.
Let $a,b,r_0,r_1$ be fixed real numbers. Then define for $ninBbb N_0$,
$$r_{n+2}=ar_{n+1}+br_n$$
and $$R(x)=sum_{ngeq0}r_nx^n$$
Multiplying both sides of the recurrence by $x^{n+2}$ and summing from $n=0$ to $infty$,
$$R(x)-r_0-r_1x=axR(x)-ar_0x+bx^2R(x)$$
$$R(x)=frac{r_0+(r_1-ar_0)x}{1-ax-bx^2}$$
Currently the most advanced thing like this I've ever done is show that
$$sum_{ngeq1}H_{n}^{(k)}s^n=frac{mathrm{Li}_k(s)}{1-s}$$
Where $$H_n^{(k)}:=sum_{i=1}^{n}frac1{i^k}$$ and $$mathrm{Li}_k(s):=sum_{ngeq1}frac{s^n}{n^k}$$ Are respectively the harmonic numbers of order $k$, and the $k$-th polylogarithm function.
Hit me!
soft-question recurrence-relations generating-functions big-list
soft-question recurrence-relations generating-functions big-list
asked Jan 17 at 5:28
clathratusclathratus
4,640337
asked Jan 17 at 5:28
clathratusclathratus
4,640337
asked Jan 17 at 5:28
clathratusclathratus
4,640337
asked Jan 17 at 5:28
clathratusclathratus
4,640337
asked Jan 17 at 5:28
clathratusclathratus
4,640337
4,640337
1
$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35
add a comment |
1
$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35
1
1
$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35
$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35
add a comment |
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$begingroup$
math.upenn.edu/%7Ewilf/DownldGF.html
$endgroup$
– Lord Shark the Unknown
Jan 17 at 5:35