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I've recently realised that recurrence relations (difference equations) can be quite powerful, especially when you start using non-constant coefficients...

  1. Can anyone recommend any good books that go deep into recurrence relations from the beginning to more complicated order equations with non-constant coefficients. Ideally with questions and answers, and reference to finance/economics.

  2. Why do university courses often have a whole course on differential equations but not an equivalent for recurrence relations. Given that they are often taught in the early stage as kind of analogous to each other, e.g. the similarity of appearance and technique of solving second-order equations...it kind of feels like they would both go on to be equally important.

My knowledge is limited so please forgive me if any of this seems trivial, thanks!

CormJack
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    I should have made the opening line..." Recently realised recurrence relations" and won the alliteration award lol – CormJack Jun 17 '22 at 15:56

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I recommend the book The Concrete Tetrahedron by M. Kauers and P. Paule. The authors put 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.

See this answer for somewhat more information.

Markus Scheuer
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  • Ah that's fantastic thank you @epi63sqrt! Do you have any thoughts on the focus on differential equations rather than recurrence relations in later studies? I googled this, but all I got was "what's the difference between them" rather than an explanation of the relative pre-eminence of differential equations. – CormJack Jun 18 '22 at 08:32
  • @CormJack: You're welcome. :-) Do you like to have a reference about differential equations or about difference equations? – Markus Scheuer Jun 18 '22 at 10:39
  • Hi! Sorry not sure I understand. I was basically asking why further studies in universities tend to have specific courses for differential equations but not for recurrence relations (difference equations)...Also is there anywhere you know I can purchase a pdf (not an ebook) of the book you recommended! – CormJack Jun 18 '22 at 10:46
  • @CormJack: Oh, yes, I see. I think recurrence relations belong to discrete mathematics which is rather a small part compared with the many mathematical disciplines where differential equations are involved. This is also to a significant part historically justified due to the strong connections e.g. between mathematics with physical investigations.. – Markus Scheuer Jun 18 '22 at 19:13
  • @CormJack: A publisher of this book is Springer, which provides his ebooks in pdf-format. – Markus Scheuer Jun 18 '22 at 19:17