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I have never been good at solving recurrence relations. Part of the reason is that I have never found a book that is good at explaining the strategies for solving them; The books just give formulas for solving recurrence relations of specific forms.

So, what books do you recommend to learn how to solve recurrence relations?

Avatrin
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    Personally, I would suggest that if you search the recurrence relations tag on this site you will find more material on theory and practical problem solving than in most books. And if there is something you don't understand it is easy to ask for clarification. – Mark Bennet Aug 07 '15 at 21:21
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    generatingfunctionology by Wilf. definitely. – gogurt Aug 07 '15 at 21:22
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    @gogurt: Yes, but that might be a bit demanding for a start; if so, the relevant parts of Graham, Knuth, and Patashnik, Concrete Mathematics, are an excellent introduction and preparation for Wilf. And Mark Bennet's suggestion is a very good one. – Brian M. Scott Aug 07 '15 at 21:32
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    @MarkBennet And if you search the entire internet you'll find even more than you can find here ;) when I buy a book I'm paying for entropy. The stuff here is usually too fragmented to use in any attempt to learn a subject. I use this site mostly as a refresher, supplement, and entertainment source, not as a collection of the most relevant material on the subject I'm interested in. – Zach466920 Aug 07 '15 at 21:43
  • @BrianM.Scott, Wilf's generatingfunctionology is not that demanding (if you skip the parts where it gets heavy, or restrict yourself to simple recurrences). It has the virtue of showing a uniform set of tools, that can be much more widely applied. And it is available for free (not in the very last edition, though) – vonbrand Aug 09 '15 at 16:24
  • @vonbrand: If someone is starting from a typical sophomore discrete math course and only modest exposure to theory-oriented math courses, Wilf is likely to be too hard. The problem isn’t so much the material as it is the pace of the presentation. The introduction in Concrete Mathematics is significantly gentler, and I found that it proved to be about as much as students in that situation could handle even in a classroom setting, not working on their own. The OP may, of course, have a better background, in which case Wilf might be just fine. – Brian M. Scott Aug 09 '15 at 16:33
  • @BrianM.Scott, personally I find "generatingfunctionology" more approachable than "Concrete Mathematics", If nothing else because its techniques don't require memorizing hundreds of abstruse identities to get anywhere. But let's agree that it is (in part) personal taste. – vonbrand Aug 09 '15 at 22:11
  • @BrianM.Scott Would you agree with vonbrand that "Concrete Mathematics" requires the memorization of hundreds of abstruse identities? – Ovi Dec 29 '16 at 02:20
  • @Ovi: No, not in the least. – Brian M. Scott Dec 31 '16 at 20:50
  • @BrianM.Scott Okay thanks, I already started Concrete Mathematics and I quite like it. Thanks for the suggestion! – Ovi Dec 31 '16 at 22:50
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    @Ovi: It has a couple of rough spots, but overall I think that it’s one of the best-written textbooks I’ve encountered in my entire career as student and teacher. You’re welcome! – Brian M. Scott Dec 31 '16 at 23:05
  • @BrianM.Scott I came across some trouble in the book, I was wondering if perhaps you could take a look? http://math.stackexchange.com/questions/2084177/maximum-number-of-regions-in-the-plane-using-zig-lines – Ovi Jan 05 '17 at 02:14
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    See Recursion Sequences by Markushevich – lhf Jul 14 '21 at 16:00

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Here I'd like to draw attention to The Concrete Tetrahedron by M. Kauers and P. Paule. This book puts the focus on four strongly connected types of mathematical objects

  • recurrences

  • generating functions

  • symbolic sums

  • asymptotic estimations

and the interplay between them. The connections and structural properties of these four regions are analysed starting with polynomials as the most simple application and going step by step, i.e. chapter by chapter to more complex objects. The authors cover

and in each of these chapters the four regions and their interplay is discussed.

Btw. the term Concrete in the title of the book is a reverence to Concrete Mathematics by R. L. Graham, D. Knuth and O. Patashnik which is explicitly stated by the authors in section 1.6:

  • The attribute concrete is a reference to the book "Concrete Mathematics" by Graham, Knuth, and Patashnik [24], where a comprehensive introduction to the subject is provided. Following the authors of that book, we understand concrete not in contrast to abstract, but as a blend of the two words con-tinuous and dis-crete, for it is both continuous and discrete mathematics that is applied when solving concrete problems.
Markus Scheuer
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RECURRENCE AND TOPOLOGY, John M. Alongi, Gail S. Nelson, Graduate Studies in Mathematics,Volume 85, American Mathematical Society. Providence, Rhode Island. Publication Year: 2007 ISBN-10: 0-8218-4234-X ISBN-13: 978-0-8218-4234-8 www.ams.org/bookpages/gsm-85

giuseppe mancò
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