The concrete tetrahedron. Symbolic sums, recurrence equations, generating functions, asymptotic estimates
This is a book on techniques for solving problems concerning infinite sequences. The authors have three goals. The first goal is to give an overview of some of the most important mathematical questions concerning infinite sequences which can be answered by a computer. The second goal is to explain how the algorithms work and on what principles they are based. The third goal is to describe some techniques which are suitable for traditional paper and pencil resoning rather than for modern computer calculations. The authors have included problems on which the techniques explained in the book can be tested. They prefer concrete examples over abstract theorems and informal explanations over formal proofs wherever this seems to be in the interest of readability, assuming basic algebraic notions such as rings, fields, etc., linear algebra in finite-dimensional vector spaces, and some complex analysis in one variable. The authors treat mathematical concepts which play a great role in many different areas of mathematics, such as symbolic sums, recurrence (difference) equations, generating functions and asymptotic estimates, and study their key features in isolation or in combination, their mastery by paper and pencil or by computer programs, and their application to problems in pure mathematics or to ``real-world problems''. These concepts are closely connected with each other: a recurrence equation has been deduced from a symbolic sum, a generating function from a recurrence and asymptotic estimates can be deduced when the generating function is available. The authors call these connections the ``concrete tetrahedron''. The book can be considered as an algorithmic supplement to [\textit{R. L. Graham}, \textit{D. E. Knuth} and \textit{O. Patashnik}, Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley Publishing Group (1994; Zbl 0836.00001)].
scientific article; zbMATH DE number 5842882