How to derive a recurrence relation from explicit form

[find_the_fun]

I am given a formula in explicit form and as a recurrence relation. It is asked to derive the recurrence relation from the explicit form. How is this done?

 

[MarkFL]

What is the closed, or explicit, form?

 

[find_the_fun]

closed: \(\displaystyle P_N=\frac{\frac{A^N}{N!}}{\sum\limits_{x=0}^N \frac{A^x}{x!}}\)

recurrence: \(\displaystyle P_i = \frac{A p_{i-1}}{i+A p_{i-1}}\) for i=1 to N

 

[MarkFL]

I was hoping you would post a closed form solution in the form:

\(\displaystyle A_n=\sum_{k=0}^m\left(c_kr_k^n\right)\)

Leading to a linear recurrence.

I will have to defer here to someone more knowledgeable in this field. :D

 

[Evgeny.Makarov]

recurrence: \(\displaystyle P_i = \frac{A p_{i-1}}{i+A p_{i-1}}\) for i=1 to N

I assume $P_{i-1}$ should be written with a capital $P$ in the right-hand side.

I'll have to think more how to derive the recurrence formula, but it is easy to prove it once it is known: just substitute the closed formula into it. It is probably easier to work with
\[
\frac{1}{P_i}=\frac{i}{AP_{i-1}}+1.
\]

 

[find_the_fun]

Can this be done using nothing but algebraic manipulation? If so, I think I found a way.