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I am reading methods of solving recurrence relation on Wikipedia. There is one method:

Many linear homogeneous recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to $$J_{n+1}=\frac{2n}{z}J_n-J_{n-1}$$ is given by $$J_n=J_n(z), \,$$ the Bessel function.

There are no description regarding how to use the method of "generalized hypergeometric series", nor can I find some on the article for generalized hypergeometric series or on Bessel function. I was wondering if someone here can explain somehow or gives some references about that? Thanks and regards!

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Recurrence relations can be solved using generating functions. Maybe this is what you are looking for? – Rudy the Reindeer Mar 6 '11 at 22:27
hypergeometric series are also mentioned in Wilf's book which is freely available here: – Rudy the Reindeer Mar 6 '11 at 22:29
@Matt: Thanks! How to solve recurrence relations using generating functions? – Tim Mar 6 '11 at 22:39
For an example (tower of hanoi) look here:… – Rudy the Reindeer Mar 7 '11 at 7:54
up vote 6 down vote accepted

See the (on-line, downloadable) book

A = B, by Petkovsek, Wilf, and Zeilberger

It gives all sorts of links between hypergeometric series and recurrence relations.

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