Consider the sequence defined by the following recurrence

Let be the gernerating function for the sequence.

Express f(z) as a rational function.

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- Oct 24th 2010, 10:04 PMsmpleasegenerating function for sequence question
Consider the sequence defined by the following recurrence

Let be the gernerating function for the sequence.

Express f(z) as a rational function. - Oct 25th 2010, 06:37 AMchisigma
The solution of the difference equation with the 'initial condition' is...

(1)

... so that is...

(2)

Now we consider separately the terms of (2)...

(3)

(4)

... so that the series (2) converges for and is...

(5)

Kind regards

- Oct 26th 2010, 04:55 AMsmplease
Thanks so much for your reply chisigma!

How did you get equation 1 though?

I have been looking for tutorials online for "difference equation" and not making much sense of it.

For eg. If its a similar question with a different sequence

How would you equate the "difference equation"

Thanks again - Oct 26th 2010, 05:15 AMPlato
Actually there is mistake in 's equation (1).

You can use this website to see the correct answer. - Oct 26th 2010, 05:33 AMsmplease
Is there a function that can be used to find this answer?

- Oct 26th 2010, 08:11 AMchisigma
A great 'Thank You' to Plato for having found my mistake... a mistake caused by my superficiality (Headbang)...

If we have a linear first order difference equation of the form...

, (1)

... its solution is...

(2)

... where . In this case the difference equation is...

, (3)

... so that is , and and the solution of (3) is...

(4)

The first term was correct and the same is for its contribution to ... the contribution of the second term has to be evaluated...

Kind regards

- Oct 26th 2010, 09:12 AMchisigma
The contribution of the second term is easy to find because is...

(1)

... so that...

(2)

Kind regards